Boulaïd Boulkroune
email: boulaid.boulkroune@u-picardie.fr

Olivier Pagès
email: opages@u-picardie.fr

Abdel Aitouche
email: abdel.aitouche@hei.fr

Ali Zemouche
email: ali.zemouche@iut-longwy.uhp-nancy.fr

Ahmed El

H ∞ -based fault diagnosis for diesel engines

The problem of fault detection for diesel engines is addressed in this work. The proposed approach allows to detect and estimate actuator and sensor faults. The residual design procedure is proposed for nonlinear parameter-varying systems. Inspired by some works in literature, the maximization problem of fault effect and the minimization problem of disturbances effect on the residuals is transformed to one problem of L 2 -norm minimization. Sufficient conditions for the existence of residual generator are formulated in terms of linear matrix inequality (LMI) sets. Performances of the proposed approach are shown through the application to a diesel engine model.

I. Introduction

In order to avoid process performance degradation, faults must be detected and located as quickly as possible in order to be able to take appropriate decisions before they turn into failure. This is why the early detection and isolation of faults is crucial for ensuring process safety and efficiency. Different fault detection and isolation (FDI) techniques are proposed in literature. The diversity of the proposed solutions was enriched by the growing interest of the industry. Among these solutions, model-based FDI systems have found extensive use due to its fast response to abrupt failure and to its implementation in real-time algorithms. This type of diagnosis method can be performed over a larger operating range. It allows also to isolate the faulty components since that a single fault in a process can propagate to several outputs. Furthermore, high diagnosis performance can be obtained despite the presence of disturbances.

Observer-based robust fault detection has received great attention in the last decades. For linear systems, various approaches have been proposed ( [START_REF] Patton | Observer-based fault detection and isolation: Robustness and applications[END_REF], [START_REF] Ding | On observer based fault detection In[END_REF] and [START_REF] Saberi | Fundamental problems in fault detection and identification[END_REF]). However, few methods are proposed for nonlinear systems and most of them are dedicated to a specific class of nonlinear systems. Besides, the extension of the existing results of linear case to nonlinear case is still a challenging task. The problem of model-based fault detection for the class of Lipschitz systems has been covered by ([4], [START_REF] Zhang | A robust detection and isolation scheme for abrupt and incipient faults in nonlinear systems[END_REF], [START_REF] Chen | Robust fault diagnosis for nonlinear difference-algebraic systems[END_REF] and [START_REF] Pertew | Lmi based sensor fault diagnosis for nonlinear lipschitz systems[END_REF]). Generally, a fault detection filter is designed in order to minimize the effect of external disturbances and to maximize its sensitivity to faults on the residuals ( [START_REF] Li | A time domain approach to robust fault detection of linear time-varying systems[END_REF]). In order to achieve these objectives, many design criteria such as H ∞ norm, H 2 norm, and H index have been proposed to evaluate the effectiveness of fault detection filter design ( [START_REF] Grenaille | In Fault diagnosis in linear parameter varying systems[END_REF], [START_REF] Ding | A unified approach to the optimization of fault detection systems[END_REF], [START_REF] Hou | An LMI approach to H /H ∞ fault detection observers In Control '96[END_REF], [START_REF] Stoustrup | Application of an H-infinity based FDI and control scheme for the three tank system[END_REF], [START_REF] Jaimoukha | A matrix factorization solution to the H /H ∞ fault detection problem[END_REF] and [START_REF] Li | A time domain approach to robust fault detection of linear time-varying systems[END_REF]). These criteria are used for constructing optimal filters : H 2 /H ∞ , H ∞ /H ∞ and H /H ∞ . The latter criterion is used to design robust fault detection filters for LPV systems (for instance [START_REF] Wei | Mixed H /H ∞ Fault Detection Observer Design for LPV Systems[END_REF] and [START_REF] Boulkroune | H /H ∞ fault detection filter for a class of nonlinear descriptor systems[END_REF]). H /H ∞ is also used in fault diagnosis for Takagi-Sugeno nonlinear systems in [START_REF] Ichalal | Fault detection, isolation and estimation for Takagie-Sugeno nonlinear systems[END_REF]. The authors are extended the method of fault diagnosis based on H ∞ control framework, developed for linear systems in [START_REF] Stoustrup | Application of an H-infinity based FDI and control scheme for the three tank system[END_REF] and [START_REF] Stoustrup | Fault estimation-a standard problem approach[END_REF] including a reference model shaping the residual signals in order to improve fault detection, isolation and estimation. Therefore, in this paper, this method will be extended to design residual generators for nonlinear parameter varying-systems with application to diesel engines.

On-board diagnosis of automotive engines has become increasingly important because of environmentally based legislative regulations such as OBDII (On-Board Diagnostics-II) [START_REF] Carb | California's obd-ii regulation (section 1968.1 title 13[END_REF]. On-board diagnosis is also needed to guarantee high-performance engine behavior. Today, due to the legislations, the majority of the code in modern engine management system is dedicated to diagnosis. Model-based diagnosis of automotive engines has been considered in earlier papers (see e.g. [START_REF] Gertler | Mode based diagnosis for automotive engines -algorithm development and testing on a production vehicle[END_REF] and [?]), to name only a few. However, the engines investigated in these previous works were all gasoline-fueled and did not include Exhaust Gas Recirculation (EGR) and Variable Nozzle Turbine (VNT). Both these components make the diagnosis problem significantly more difficult since the air flow through the EGR-valve, and also the exhaust side of the engine have to be taken into account. An interesting approach to model-based air-path faults detection for an engine which includes EGR and VNT can be found in [START_REF] Nyberg | Model-based diagnosis of an automotive engine using several types of fault models[END_REF] and [START_REF] Nyberg | Model based diagnosis of the air path of an automotive diesel engine[END_REF]. By using several models in parallel, where each one is sensitive to one kind of fault, predicted outputs are compared and a diagnosis is provided. In particular, the hypothesis test methodology proposed in [START_REF] Nyberg | Model-based diagnosis of an automotive engine using several types of fault models[END_REF] deals with the multi-fault detection in air-path system. In [START_REF] Nyberg | Model based diagnosis of the air path of an automotive diesel engine[END_REF] the authors propose an extended adaptive Kalman filter to find which faulty model best matches with measured data, then a structured hypothesis allows going back to the faults. A structural analysis for air path of an automotive diesel engine has been developed in order to study the monitorability of the system [START_REF] Djemili | Structural analysis for air path of an automotive diesel engine[END_REF]. Other approaches to detect intake leakages in diesel engines based on adaptive observers are proposed in [START_REF] Ceccarelli | Model-based adaptive observers for intake leakage detection in diesel engines[END_REF] and [START_REF] Ceccarelli | Robust strategy for intake leakage detection in diesel engines[END_REF] and recently in [START_REF] Djemili | Adaptive observer for intake leakage detection in diesel engines described by takagi-sugeno model[END_REF]. Another approach based on a nonlinear unknown input observer (NIUO) for intake leakage detection is proposed in [START_REF] Boulkroune | Nonlinear unknown input observer design for diesel engines[END_REF]. For sensor fault diagnosis, few results have been proposed in the literature although the use of healthy measurements is strongly required in such systems. In [START_REF] Capriglione | On-line sensor fault detection, isolation, and accommodation in automotive engines[END_REF], on-line sensor fault detection, isolation, and accommodation in automotive engines was proposed. In [START_REF] Kimmich | Fault detection for modern diesel engines using signal-and process modelbased methods[END_REF], fault detection for diesel engines using signal and process model-based methods was proposed. An expert system for fault diagnosis system in internal combustion engines using wavelet packet transform (WPT) and artificial neural networks (ANNs) techniques has been proposed by [START_REF] Wu | An expert system for fault diagnosis in internal combustion engines using wavelet packet transform and neural network[END_REF]. Recently, a sensor fault detection approach for diesel engines is proposed in [START_REF] Boulkroune | Sensor fault detection for diesel engines[END_REF]. This approach is based on nonlinear unknown input observers which are used to estimate jointly the original states and the sensor fault. This work deals with the problem of fault detection for diesel engines. The proposed approach can be used for detecting, isolating and estimating actuator and sensor faults in nonlinear parameter-varying systems. First, the nonlinear structure of the system is used in the residual generation to keep the system dynamics. Based on the the modified mean value theorem (MMVT) presented in [START_REF] Phanomchoeng | Nonlinear observer for bounded jacobian systems, with applications to automotive slip angle estimation[END_REF], the nonlinear term in the error dynamics is expressed as a convex combination of known matrices with time varying coefficients. Then, the problem of disturbance effect minimization and fault effect maximization on residuals is reduced to a problem of L 2 -norm minimization. This was possible by building the residual generators as a difference between real residuals, i. e. difference between real output and its estimate, and a reference model virtually fed by the fault signal. The reference model is chosen as first order low-pass filter. Finally, the L 2 -norm minimization problem is solved by applying the bounded real lemma (BRL). The main advantage of this method is its applicability to a wide large operating points and the residual generator gains are determined off-line which making its real online application very easy.

II. Preliminaries and definitions

We introduce some notation and definitions that will be used throughout this paper. ℜ denotes the set of real numbers. The set of p by q real matrices is denoted as ℜ p×q . A T and A -1 represent the transpose of A and its left inverse (assuming A has full column rank), respectively. I r represents the identity matrix of dimension r. (⋆) is used for the blocks induced by symmetry. . represents the usual Euclidean norm.

Let us consider the following class of nonlinear systems

ẋ = A(ρ j )x+B g g(υ, y, u)+B f f (x, u)+B d d+B w w (1a) y = Cx + D d d + D w w (1b)
with A(ρ j ) = ∑ n ρ j=1 ρ j A j , where x ∈ ℜ n x is the state vector, u ∈ ℜ n u is the control input vector, y ∈ ℜ n y is the output vector, υ ∈ ℜ n υ is the vector of measurable signals, d ∈ ℜ n d represents the actuator and sensor fault vector and w ∈ ℜ n w is the disturbances vector. A j , with j = 1,

• • • , n ρ , B g , B f , B d , B w , D d , D w and C are constant matrices with appropriate dimensions. All the pairs (A i , C), with i = 1, • • • , n ρ ,
are assumed detectable. Functions g (υ, y, u) and f (x, u) are nonlinear. Besides, function f (x, u) is assumed to be once differentiable. The weighting functions ρ j are assumed known and depend on measurable variables, and verify

n ρ ∑ j=1 ρ j = 1, ρ j ≥ 0, ∀ j ∈ {1, • • • , n ρ } (2)
It is important to note that the structure of system (1) is different from Takagi-Sugeno (TS) models. In fact, sometimes it is not important to write nonlinear models in a TS form especially when the number of nonlinearities is large as the case of diesel engine models (see section IV). Consequently, rewrite nonlinear models as in ( 1) becomes an interesting solution to overcome these difficulties. Besides, most of nonlinear systems can be easily written as in (1) without any transformation.

For manipulating the nonlinear function f , the MMVT theorem presented for a general vector function f in [START_REF] Phanomchoeng | Nonlinear observer for bounded jacobian systems, with applications to automotive slip angle estimation[END_REF] is applied in this work. It is used to express the nonlinear error dynamics as a convex combination of known matrices with time varying coefficients. The main principle of MMVT is presented in the following Lemma.

Lemma 1: [ [START_REF] Phanomchoeng | Nonlinear observer for bounded jacobian systems, with applications to automotive slip angle estimation[END_REF]] Let the canonical basis of the vectorial spase ℜ s for all s ≥ 1 be defined by

E s = {e s (i) | e s (i) = 0, ..., 0, i th 1 , 0, ..., 0 s components T , i = 1 • • • s} Let f (x) : ℜ n → ℜ n be a vector function continuous on [a, b] ∈ ℜ n and differentiable on convex hull of the set (a, b). For s 1 , s 2 ∈ [a, b], there exist δ max i j and δ min i j for i = 1 • • • n and j = 1 • • • n such that f (s 2 )-f (s 1 ) = n,n ∑ i, j=1 H max i j δ max i j + n,n ∑ i, j=1
H min i j δ min i j

(s 2 -s 1 )

δ max i j , δ min i j ≥ 0, δ max i j + δ min i j = 1 (3) • h max i j ≥ max( ∂ f i ∂ x j
) and

h min i j ≤ min( ∂ f i ∂ x j ) • H max i j = e n (i)e T n ( j)h max i j
and H min i j = e n (i)e T n ( j)h min i j .

III. Observer-based residual generation design

The aim of this work is to design a residual generator for nonlinear parameter-varying systems described by [START_REF] Patton | Observer-based fault detection and isolation: Robustness and applications[END_REF]. Indeed, the system (1) represents a large class of nonlinear systems. Let's consider the following residual filter :

ẋ = A(ρ) x+B G g(υ, y, u)+B f f ( x, u)+L(ρ)(y-ŷ) (4a) ŷ = Cx (4b) r = K(y-ŷ) (4c)
with L(ρ) = ∑ n ρ j=1 ρ j L j . x ∈ ℜ n x and r(t) ∈ ℜ n r are respectively the estimated state vector and the residual vector. The residual generator gains are matrices L and K. These matrices should be determined such that the asymptotic stability of residuals is guaranteed. Notice that the index ρ is omitted where it is not necessary in order to simplify the notations.

First, let define the state estimation error e as e(t) = x(t)x(t). Then, the error dynamics and residual expression can be expressed as:

ė = (A-LC)e+B f f +(B d -LD d )d+(B w -LD w )w (5a) r = KCe+KD d d+KD w w (5b) with f = f ( x, u) -f (x, u).
Now, by using Lemma 1, the nonlinear term f ( x, x) can be written as

f ( x, x) = n,n ∑ i, j=1 H max i j δ max i j + n,n ∑ i, j=1
H min i j δ min i j e (6a)

δ max i j , δ min i j ≥ 0, δ max i j + δ min i j = 1 (6b)
To simplify the form of the final result as in [START_REF] Phanomchoeng | Nonlinear observer for bounded jacobian systems, with applications to automotive slip angle estimation[END_REF], the terms

n,n ∑ i, j=1
(δ max i j + δ min i j ) must be scaled to one. If all the terms in ∂ f i ∂ x j are not zero, then the scaling factor β can be computed by n,n ∑ i, j=1

δ max i j + δ min i j = n 2 = β .
Then, (6a) are rewritten as f = αe, where α =

n,n ∑ i, j=1 Hmax i j δ max i j + n,n ∑ i, j=1 Hmin i j δ min i j
, with

δ max i j , δ min i j ≥ 0, δ max i j + δ min i j = 1 β , n,n ∑ i, j=1 δ max i j + δ min i j = 1
where Hmax

i j = β H max i j , δ max i j = δ max i j /β and δ min i j = δ min i j /β .
Then, (5) becomes : As seen in equation ( 7), the error dynamics and the residual r depend only on the fault d and the disturbances w. Now, the fault detection problem is reduced to minimize the effect of disturbances and to maximize the effect of the fault on the residual simultaneously. Indeed, one solution is to use the well known H /H ∞ approach. Thus, the effect of disturbances on the residuals is minimized using the H ∞ norm while the effect of the fault on the residual is maximized using the H index. However, using the two performances simultaneously increases the problem complexity. In this work, the H /H ∞ optimization problem is reduced to one H ∞ optimization problem by introducing the following variable r e expressed as :

r e = r -W f d (8)
where W f is a transfer function representing the desired transfer from the fault d to residual r and is called reference model. The idea of a reference model has successfully used in controller design, adaptive control and diagnosis. Therefore, the maximization of the effect of the fault on the residual r is equivalent to minimization of the effect of the fault on the residual error r e .

As explained in [START_REF] Stoustrup | Application of an H-infinity based FDI and control scheme for the three tank system[END_REF], structure of W f can be chosen depending on the fault diagnosis purpose. Thus, the fault estimation problem can be obtained by choosing W f equal to the identity matrix, W f = I and the fault detection problem can be obtained when W f is a 1 × n d matrix. Also, W f is not required to be a static matrix. In this work, W f is chosen as

W f = A f B f C f D f (9)
Now, from ( 7) and ( 8), the following augmented system is obtained :

ξ = A ξ ξ + B ξ ν (10a
)

r e = Cξ ξ + Dξ ν (10b)
where

ξ = e x f , ν = d w , (11a)
A ξ = A e 0 0 A f , B ξ = B ed B ew B f 0 (11b) Cξ = C e -C f , Dξ = D rd -D f D rw (11c)
The problem now is reduced to determine the gains L and K that provides convergence of ξ and minimizes the influence of the fault and disturbances on the residual error (r e ) using the standard L 2 -control problem. In this case, r e represents the controlled output. It is worth to note that in the fault estimation problem, the transfer function from the vector ν to the residual r e includes a direct term D ξ which is not null. Consequently, the lower bound (γ min ) of H ∞ norm of the closed loop transfer function from internal input to external output will be greater than or equal to D ξ ∞ . When D ξ = I, γ min will be greater than or equal to 1 which leads to an over estimation of the fault estimation error. To overcome this difficulty, two solutions are proposed in [START_REF] Stoustrup | Application of an H-infinity based FDI and control scheme for the three tank system[END_REF]. The first solution is to filter the estimation error using a low-pass filter by taking : r e f = W re r e [START_REF] Stoustrup | Application of an H-infinity based FDI and control scheme for the three tank system[END_REF] where W re is a low pass weighting matrix. The second solution is to model the fault signal as the output signal from a low pass dynamic system by taking

d = W d d ( 13
)
where W d is a diagonal weighting matrix with first-order low pass transfer functions in the diagonal. The first approach is used in this work with a constant matrix W re .

Based on ( 13), the augmented system ( 10) is rewritten as follows

ξ = A ξ ξ + B ξ ν (14a) r e f = C ξ ξ + D ξ ν (14b)
where

C ξ = W re C e -C f , (15a)
D ξ = W re D rd -D f D rw (15b)
The sufficient condition of the existence of the residual generator r is given in theorem 1.

Theorem 1: System (14a)-( 14b) is asymptotically stable if there exist a symmetric positive definite matrix P > 0 and a positive scalar γ > 0 such that

  Λ max i jk P B -R k D ( K C + Cf ) T ( * ) -γI ( K D + D f ) ( * ) ( * ) -γI   < 0 (16a)   Λ min i jk P B -R k D ( K C + Cf ) T ( * ) -γI ( K D + D f ) ( * ) ( * ) -γI   < 0 (16b) ∀i = 1, • • • , n, j = 1, • • • , n and k = 1, • • • , n ρ , with • Λ max i jk = (A k + B f Hmax i j ) T P + P(A k + B f Hmax i j ) -CT R T k -R k C; • Λ min i jk = (A k + B f Hmin i j ) T P + P(A k + B f Hmin i j ) -CT R T k -R k C;
where Hmax

i j = Z H H max i j , Hmin i j = Z H H min i j , with Z H = n × n. Matrices B, C, D, Cf , D f and K are given as B = B d B w B f 0 , C = C 0 , (17a)
D = D f D w , Cf = W re 0 -C f , (17b) D f = W re -D f 0 , K = W re K (17c)
Matrices P, R k and K are obtained by solving LMIs (16a)-(16b). The matrices L k can be obtained from L k = I 0 P -1 R k .

Proof 1: Based on linear bounded lemma [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF], the transfer from ξ to r e is minimized using H ∞ criterion if a matrix P and a positive constant γ exist such that the following condition holds:

Ξ =   A T ξ P + PA ξ PB ξ C T ξ ( * ) -γI D ξ ( * ) ( * ) -γI   < 0 ( 18
)
where P is positive symmetric matrix. From (11b), (15a) and (15b), ( 18) can be expressed as

Ξ =   ĀT P + P Ā -CT R T -R C P B -R D ( K C + Cf ) T ( * ) -γI ( K D + D f ) ( * ) ( * ) -γI   where Ā = A + B f α, R = P L 0 . Matrices B, C, D, Cf ,
D f and K are given in [START_REF] Ichalal | Fault detection, isolation and estimation for Takagie-Sugeno nonlinear systems[END_REF].

By substituting α by its expression, we obtain :

Ξ = n,n ∑ i, j=1 δ max i j   Λ max i j P B -R D ( K C + Cf ) T ( * ) -γI ( K D + D f ) ( * ) ( * ) -γI   M max i j + n,n ∑ i, j=1 δ max i j   Λ min i j P B -R D ( K C + Cf ) T ( * ) -γI ( K D + D f ) ( * ) ( * ) -γI   M min i j with Λ max i j = (A + B f Hmax i j ) T P + P(A + B f Hmax i j ) -CT R T -R C, Λ min i j = (A + B f Hmin i j ) T P + P(A + B f Hmin i j ) -CT R T -R C.
Since we have δ max i j , δ min i j ≥ 0, the negativity of Ξ is ensured if

M max i j < 0 and M min i j < 0 (19)
By substituting A(ρ) and L(ρ) by their expressions, [START_REF] Gertler | Mode based diagnosis for automotive engines -algorithm development and testing on a production vehicle[END_REF] can be written as follows:

n ρ ∑ k=1 ρ k M max i jk < 0, n ρ ∑ k=1 ρ k M min i jk < 0 for all k = 1, . . . , n ρ (20)
where

M max i jk =   Λ max i jk P B -R k D ( K C + Cf ) T ( * ) -γI ( K D + D f ) ( * ) ( * ) -γI   M min i jk =   Λ min i jk P B -R k D ( K C + Cf ) T ( * ) -γI ( K D + D f ) ( * ) ( * ) -γI   with • Λ max i jk = (A k + B f Hmax i j ) T P + P(A k + B f Hmax i j ) -CT R T k -R k C; • Λ min i jk = (A k + B f Hmin i j ) T P + P(A k + B f Hmin i j ) -CT R T k -R k C; Since ρ k > 0, ∀k = 1, • • • , n ρ , the condition (20) holds if M max i jk < 0 and M min i jk < 0, ∀i = 1, • • • , n, j = 1, • • • , n and k = 1, • • • , n ρ .
This ends the proof.

IV. Engine model

The diesel engine considered in this paper is a four-cylinder engine with a EGR and a VNT. Due to space limitation, we present directly the air-path model writing in state space form as [START_REF] Patton | Observer-based fault detection and isolation: Robustness and applications[END_REF]. For more details we refer the reader to [START_REF] Boulkroune | Sensor fault detection for diesel engines[END_REF]. The engine is equipped with sensors measuring in-flowing air W HFM , the temperature after CAC T CAC , inlet-manifold pressure P Inlet , exhaustpressure P Exh and exhaust-temperature T Exh . The control inputs are the injected fuel W Fuel , the turbine vane position X V NT , and the EGR-valve position determining the valve opening-area A EGR . Notice that the following variables : T CAC , W HFM , and N Eng are considered as measurable signals.

The considered air path model is expressed as follows [START_REF] Patton | Observer-based fault detection and isolation: Robustness and applications[END_REF], where the state, known input and output vectors and the variables ρ and υ are defined as

x = P Inlet m Air m EGR m Exh T (21) u = A EGR X V NT W Fuel T (22)
y = P Inlet P Exh T (23) ρ = N Eng (24) υ = T CAC W HFM T ( 25
)
where m Air and m EGR are respectively the mass of air and EGR-gaz in intake manifold. m Exh represents the mass of exhaust gas in exhaust manifold. The matrices A 1 and A 2 are given by :

A 1 =    -a 1 0 0 0 0 -a 1 0 0 0 0 -a 1 0 0 a 1 a 1 0    , A 2 =    -a 2 0 0 0 0 -a 2 0 0 0 0 -a 2 0 0 a 2 a 2 0    with a 1 = f vol N Eng V Eng 120V Inlet and a 2 = f vol N Eng V Eng 120V Inlet
, where N Eng and N Eng are, respectively, the minimum and maximum value of the measurable variable N Eng . ρ 1 and ρ 2 are defined as

ρ 1 = N Eng -N Eng N Eng -N Eng and ρ 2 = N Eng -N Eng N Eng -N Eng (26)
Matrices B g , B f , B d , B w , D w , C and D d are expressed as :

B g =     R Air c p,Air c v,Air V Inlet 0 0 1 0 0 0 0     , B f =       - f vol V Eng N Eng 120V Inlet R 2 Exh T EGR c p,Exh V Inlet c v,Exh V Exh 0 0 0 0 0 R Exh V Exh 0 0 -R Exh V Exh 1       , (27)
B d =    0 0 0 0 0 0 0 0 0 0 0 13 × 10 -4    , B w =    0 0 0 0 0 0 0 0    , D w = 1 0 0 1 (28) C= 1 0 0 0 0 0 0 R Exh T Exh V Exh , D d = 2 × 10 4 0 0 0 2.5 × 10 4 0 (29)
Functions g and f are given by :

g = W HFM T CAC W HFM , f =      R Inlet P Inlet c v,Inlet A EGR m Exh Ψκ Exh T Exh √ R Exh T Exh W Fuel - R Exh T Exh m Exh τ P Exh P Atm ,X V NT V Exh √ T Exh     
All the parameters and variables used in this model are listed in Table in appendix ??. The temperature T EGR and T Exh are assumed to be constant. The static functions : f vol , h and τ are represented as interpolation in lookup tables.

V. Simulation results

The proposed approach has been tested on a fourcylinder diesel engine model running on Mathworks Matlab /Simulink. The aim is to detect two types of faults : actuator and sensor faults. The two first components of the vector d are, respectively, the inlet and exhaust manifold sensor faults. Two step-like faults with a magnitude equal to 10% of the mean values of inlet and exhaust pressures are considered. The last component on the fault vector is an actuator fault represented by a step-like fault with a magnitude equal to 40% of the mass-flow of the injected fuel. This fault is due either to a drop in the pressure in the common rail or a dysfunction of one or more injectors.

Due to space limitations, the numerical values of vectors and matrices, in Theorem 1, are omitted. Notice that the LMIs (16a)-(16b) are solved using YALMIP, a toolbox for modeling and optimization in Matlab . In this work, a dedicated residual for each fault is constructed. Besides, this structure helps us to isolate easily each fault. In addition, in each case W f is chosen as a stable first order low-pass filter. By solving the LMIs (16a)-(16b), the following optimal values of γ are obtained: γ * 1 = 0.8665, γ * 2 = 0.9083 and γ * 2 = 0.7141.

The simulation is performed with engine average speed N Eng = 3000rpm (see Fig. 1). In this experience, the minimum and maximum value of N eng (N eng and N eng ) are chosen as : 2000 and 4000rpm respectively. For simulation purpose, we consider that the measurements are perturbed by a gaussian white noise with a standard deviation equal to 5 × 10 3 . N eng times (s)

VI. Conclusion

The design of residual generators for nonlinear parameter-varying systems is treated in this paper. The proposed approach aims to detect actuator and sensor faults in diesel engines. The disturbance effect minimization and fault effect maximization problem is transformed to a problem of L 2 -norm minimization. The residual generator gains are obtained by solving a set of LMIs. The efficiency of the proposed method is shown through the application to a diesel engine model where the effect of each fault is well shown on the equivalent residual.

ė

  = A e e + B ed d + B ew w (7a) r = C e e + D rd d + D rw w (7b) with A e = A -LC + B f α, B ed = B d -LD d , B ew = B w -LD w , C e = KC, D rd = KD d and D rw = KD w .

Fig. 1 .

 1 Fig. 1. Engine speed behavior

Fig. 2 .Fig. 3 .

 23 Fig. 2. Inlet manifold pressure sensor fault

ACKNOWLEDGMENTS

This work was produced in the framework of CEREEV (Combustion Engine for Range-Extended Electric Vehicle), a European territorial cooperation project part-funded by the European Regional Development Fund (ERDF) through the INTERREG IVA France (Channel)-England program, and the research department of the Picardie, France.


Guillaume Dessevre

Pierre Baptiste

Jacques Lamothe

Robert Pellerin

Visual charts produced by simulation to correlate service rate, resource utilization and DDMRP parameters

Keywords: Decision support, capacity management, decoupled lead time, DDMRP, simulation

Demand Driven Material Requirements Planning (DDMRP) is a recent method mixing push and pull flow management. Although it claims to be the solution to traditional methods' limitations, the DDMRP method works at infinite capacity: manufacturing or supply orders are launched according to a logic of replenishment of stocks defined as buffers. This article proposes an evaluation of capacity management using visual charts developed by simulation. These charts correlate the bottleneck resource's loading rate to a service rate by considering one of the DDMRP method parameters, the Decoupled Lead Time (DLT). The charts are a decision support tool. They allow identifying to which loading rate the DLTs are representative of the flow times of manufacturing orders and which capacity level to use. We study different workshops, including a real industrial case. Our results show that it is better to control the flow times by adjusting capacity rather than adjust the DLT parameter.

Introduction

Demand Driven Material Requirements Planning (DDMRP) is a production planning and management method introduced by [START_REF] Ptak | Orlicky's Material Requirements Planning 3/E[END_REF]. In integrating push and pull flows, this method is based on buffers' strategic positioning along the Bill of Material (BOM). DDMRP is one of the production control systems that emerged recently [START_REF] Bagni | Systematic Review and Discussion of Production Control Systems That Emerged Between 1999 and 2018[END_REF].

As shown in Figure 1, each stock buffer is sized according to the Average Daily Demand (ADU), which can change over time, and several parameters, including the Decoupled Lead Time (DLT), which is defined as the longest upstream lead time not protected by a buffer, the Lead Time Factor (LTF), the Variability Factor (VF) and the Minimum Order Quantity (MOQ).

A replenishment order to the TopOfGreen level is placed anytime a TopOfYellow threshold is exceeded (Equation 1):

Physical inventory + Work -in -process -(Day demand + Peak demand) ≤ TopOfYellow (1)
A peak is an original concept of DDMRP. It corresponds to an exceptionally large demand in the backlog over a given time horizon [START_REF] Ptak | Demand Driven Material Requirements Planning (DDMRP)[END_REF].

guillaume.dessevre@polymtl.ca Département de Mathématiques et de Génie Industriel, Polytechnique Montréal, CONTACT Guillaume Dessevre Montréal, Canada It generates a replenishment on demand instead of on consumption. Dessevre et al. (2019) have shown that the choice of DLT can influence shop floor performance, including customer service rate. However, the 'DLT -service rate' relationship still deserves to be clarified. [START_REF] Hopp | Factory Physics: Foundations of Factory Management[END_REF] define a shop floor's service rate as the probability that the order flow times are less than or equal to the allocated lead time. In a DDMRP context, the shop service rate can be defined as the probability that these flow times are less than or equal to the DLT (Equation 2).

Workshop Service Rate = P{ Flow Time ≤ DLT} (2) Therefore, the DLT is an important parameter linked to both the service rate and the stock level (because it determines the buffer's zones): a too small DLT will not absorb all the flow times and may cause shortages, degrading both customer service rate and workshop service rate, and a too larger DLT will increase stock levels and the associated costs.

Also, queuing theory shows that flow times in production lines increase drastically when the loading rate increases [START_REF] Kingman | Some Inequalities for the Queue GI/G/1[END_REF]). However, DDMRP operates at infinite capacity, suggesting that scrolling times in a DDMRP managed shop can increase rapidly as the loading rate increases. To answer the capacity issue, [START_REF] Ptak | Orlicky's Material Requirements Planning 3/E[END_REF] propose a capacity buffer defined as 'the protective capacity at both constraint and unconstraint resources that allows these resources to catch up when Murphy strikes', referring to Murphy's law: what can go wrong will go wrong. Unfortunately, a capacity buffer is just an additional amount of capacity to protect the system: it gives no recommendation, and it may be incorrectly sized.

This paper aims to create visual charts that correlate the bottleneck resource loading rate, the workshop service rate, and the DLT parameter by simulating different DDMRP managed workshops, including an industrial case.

The created charts will help to answer the following questions:

• Up to what loading rate of the bottleneck resource can the DLT absorb flow times? • What percentage of flow time is less than or equal to this DLT? • How does the service rate behave according to the loading rate? • What can a production manager do to get under control the flow times and so the service rate?

Answering these questions will allow a better parameterisation of the DDMRP method, as it will provide an assessment of the sizing of the DLT, with a visualisation of the parameterisation choices' consequences. Moreover, our charts can be used to determine the size of the capacity buffer. This paper is an extension of a previous one Dessevre, Baptiste, and Lamothe (2020), where we go further in the reasoning: we study here a real industrial case, we compare it with our fictional workshop, we answer another management problem (the choice of the number of shifts that the production manager must decide, explaining how to use the charts) and we show limits of the charts. The two papers are complementary in this research project.

This article is organised as follows. First, Section 2 presents a review of the literature on publications related to the topic. Section 3 then describes the research methodology (the workshops studied, the experimental design, and the simulation parameters). Section 4 presents the results, and finally, Section 5 concludes and proposes avenues for further research.

Literature review

First, The DDMRP method is a controversial subject. Often restricted to a 'consultants' method', it has made its way into the academic world with an increasing number of research articles dedicated to it. The first publications have demonstrated the method's relevance by comparing it to other traditional MRPII and Kanban methods, showing the force of the DDMRP: a better compromise between stock level and service rate, the anticipation of peak demand, dynamic adjustment of buffer sizing, and the ability to work with high product diversity [START_REF] Ihme | Evaluating Demand Driven MRP: a Case Based Simulated Study[END_REF][START_REF] Miclo | An Empirical Comparison of MRPII and Demand-Driven MRP[END_REF][START_REF] Shofa | Effective Production Control in an Automotive Industry: MRP vs. Demand-Driven MRP[END_REF][START_REF] Miclo | Demand Driven MRP: Assessment of a new Approach to Materials Management[END_REF]. Today, the field of research on DDMRP has expanded. While some are interested in its strategic perspective, like [START_REF] Vidal | Toward an Aggregate Approach for Supporting Adaptive Sales And Operations Planning[END_REF] studying the Adaptive Sales & Operations Planning, others focus on the mechanics and operational parameters. For instance, [START_REF] Martin | A Process map for the Demand Driven Adaptive Enterprise Model: Towards an Explicit Cartography[END_REF] propose a decision tree allowing a better parameterisation of buffers. Dessevre et al. (2019) are interested in DLT and LTF parameters by putting them under control. Recently [START_REF] Lee | A Mathematical Safety Stock Model for DDMRP Inventory Replenishment[END_REF] propose an alternative to the safety stock calculation model. The comparison of DDMRP with other methods is still up to date, as [START_REF] Thürer | Production Planning and Control in Multi-Stage Assembly Systems: an Assessment of Kanban, MRP, OPT (DBR) and DDMRP by Simulation[END_REF] compare four production control systems, showing the potential of DDMRP in multi-stage assembly systems. Nowadays, studies on DDMRP are both axiomatic and empirical [START_REF] Bagni | Systematic Review and Discussion of Production Control Systems That Emerged Between 1999 and 2018[END_REF]. However, there are still many issues to be addressed scientifically, while more and more companies are developing DDMRP in many industrial sectors [START_REF] Bahu | Compréhension du DDMRP et de son Adoption: Premiers éléments Empiriques[END_REF]. Therefore, researchers aim to study the method in more complex environments [START_REF] Acosta | Applicability of Demand-Driven MRP in a Complex Manufacturing Environment[END_REF], raising new questioning from particular industrial sectors (Dessevre et al. 2020), and bringing the need of a standardised implementation process for the method DDMRP [START_REF] Orue | Demand Driven MRP-The Need to Standardise an Implementation Process[END_REF].

Then, we focus on the role of lead time in a workshop in literature. [START_REF] Hopp | Factory Physics: Foundations of Factory Management[END_REF] define 'the lead time of a given routine or line is the time allotted for production of a part on that routeing or line'. They clearly explain that lead time is different from flow time, as the former is a management choice, and the latter is generally random. A good lead time must absorb flow times and their variations, hence Equation 2, but it must but as small as possible to limit stocks: in a DDMRP context for example, the greater the lead time, the greater the DLT, and therefore the greater the stock buffers (Figure 1). Moreover, [START_REF] Christensen | Variance vs Average: Supply Chain Lead-Time as a Predictor of Financial Performance[END_REF] show that lead time is linked to financial performance. That is why a branch in the literature is about 'controllable' times, where these studies focus on reducing lead times in procurement (preparation and transport time) and/or in production (changeover time, production time, speed, batch size, etc.). For example, [START_REF] Sarkar | Quality Improvement and Backorder Price Discount Under Controllable Lead Time in an Inventory Model[END_REF] studied two models with different demand distributions, [START_REF] Jha | Single-vendor Multi-Buyer Integrated Production-Inventory Model with Controllable Lead Time and Service Level Constraints[END_REF] included a constraint on the service rate in their model, [START_REF] Glock | Lead Time Reduction Strategies in a Single-Vendor-Single-Buyer Integrated Inventory Model with lot Size-Dependent Lead Times and Stochastic Demand[END_REF] proposed methods for reducing lead time in a single-vendor-single-buyer model, and many others [START_REF] Pan | A Study of an Integrated Inventory with Controllable Lead Time[END_REF][START_REF] Ouyang | Integrated Vendor-Buyer Cooperative Models with Stochastic Demand in Controllable Lead Time[END_REF][START_REF] Hidayat | Supplier Selection Model Development for Modular Product with Substitutability and Controllable Lead Time[END_REF][START_REF] Shin | Controllable Lead Time, Service Level Constraint, and Transportation Discounts in a Continuous Review Inventory Model[END_REF]). Thus, many scientists consider lead times as a decision variable and try to size it as best as possible.

Finally, the factor ρ/(1-ρ), where ρ represents the loading rate, is present in the formulas for calculating average flow times from the queue theory of 1-server systems: [START_REF] Kingman | Some Inequalities for the Queue GI/G/1[END_REF], [START_REF] Marchal | An Approximate Formula for Waiting Time in Single Server Queues[END_REF] or [START_REF] Krämer | Approximate Formulae for the Delay in the Queueing System GI/G/l[END_REF]. Therefore, the higher the loading rate, the more drastically the times increase. To avoid this issue, it is important to control the loading rate, especially for the bottleneck resource. [START_REF] Ptak | Orlicky's Material Requirements Planning 3/E[END_REF] introduce a capacity buffer, which is an additional amount of capacity in order to absorb variability. They explain that 'capacity buffers are not being used [. . . ] to maximise a resource's utilisation or efficiency. [They] require that a resource maintain a bank a capacity that goes unused' [START_REF] Ptak | Demand Driven Material Requirements Planning (DDMRP)[END_REF]. But how to determine the size of that bank? And if it is not enough? They answer the former by analyzing the demand variability, and the latter by proposing different long-term methods such as reengineering the products to manipulate the load or raising price to manipulate the demand. Thereby, there is no answer for short-term capacity issues.

As a conclusion of this review, it is known that (i) flow times are correlated with the loading rate (by the queue theory), that (ii) lead time is a decision variable that can be reduced at a certain cost (controllable times), that (iii) the DLT parameter of the DDMRP method is used in the dimensioning of buffers and that (iv) it is an important parameter related to the service rate of the workshop. One question thus arises: How can we control a service rate in a DDMRP-managed shop subject to load variations? To answer this question, we propose to create charts correlating the service rate, the loading rate, and the DLT parameter.

Methodology

Our research strategy is based on the study of two simulated cases: a fictional flowshop and an industrial case. Both cases are similar: products are buffered and manufactured in a workshop where components are buffered too. In this way, there is always a flowshop with a bottleneck station between two buffers. If both cases give similar results, we might work on the fictional case in future research between two buffers, as it is easier to change fictional parameters (number of stages, products, etc.). Simulation has been chosen because it easily allows to model and analyze complex environments with several sources of variability [START_REF] Mourtzis | Simulation in the Design and Operation of Manufacturing Systems: State of the art and new Trends[END_REF]. Furthermore, the objective is not to optimise but to observe a phenomenon, it is therefore not necessary to develop more 'fine' tools.

This section describes the workshops studied as well as the simulation parameters used.

The ictional lowshop

Workshop parameters

The workshop studied is a production line composed of 6 workstations (one machine per workstation) with DDMRP stock buffers at the beginning and end of the line (for components and finished products, as shown in Figure 2). This type of workshop was chosen since it corresponds to what can be found between two DDMRP stock buffers in the industry.

Since DDMRP performs well in highly variable environments, several sources of variability were introduced into the system (demand, production, etc.). The first machine is subject to 4-hour outages every 36 h and the Thirty products are manufactured in the workshop. Series changeover times (in hours) and production times (in seconds per part) per machine are presented in Tables 1 and2. To simulate variability in production times, the changeover times are multiplied by a variable following a triangular law (0.75, 1, 1.25) and the production times by a variable following a triangular law (0.8, 1, 1.2). An operator is required during changeover and production operations for a machine to operate.

Each product has a different average order size, and this order size follows a uniform law of ±20% around the average as an arbitrary choice after observing demand signals from the industrial case. The delay between each order of the same product follows an exponential law of expectation of 1 d. Therefore, the Average Daily Usage (ADU) of each product equals the average order size, shown in Table 3.

The DLT, LTF, and VF are equal for each product and are valid for 10 days, 50%, and 50%, respectively. This choice of parameterisation comes from previous research conclusions (Dessevre et al. 2019).

The workshop and operators work 8 h a day, 5 days a week. Since the components are themselves managed on DDMRP stock buffers, they are considered available at all times in sufficient quantities. Finished product stocks are randomly initialised between 50% and 100% of the Top Green of each buffer.

Because of breakdowns and product mix, the bottleneck is globally located on the M5 machine but can temporarily shift to other machines. Moreover, when the number of operators is reduced, the bottleneck resource becomes the operators.

The workshop manages partial orders: when a customer order arrives, it is delivered in full if possible. Otherwise, it enters a queue and will be given priority when the product's stock in question is available again.

Design of experiments and simulation parameters

For the fictional case, the goals of the design of experiments are:

• To verify the average flow time of all production orders represents the average flow time of productions orders for each finished product; • To compare a case where the bottleneck resource is a machine and the one where it is the operators; and • To create flow charts that will be compared with the industrial case.

The input and output variables of the design of experiments are presented here. There are two input variables: the average customer demand and the number of operators.

To generate a progressive scale-up, the average size of each order has been uniformly increased step by step. The aim is to have a load/capacity ratio between 70% and 100% (in all demand scenarios, below this ratio, the workshop has overcapacity, above this ratio, it is overused and saturates very quickly).

Also, two cases are studied to test two different types of critical resources:

• Case 1: the bottleneck is a machine (the fifth on the line) and 6 operators are present in the workshop; and • Case 2: 5 operators are present in the workshop and represent the bottleneck resource.

The output variables of the experimental design are:

• The loading rate of the bottleneck resource;

• The workshop service rate, defined according to [START_REF] Hopp | Factory Physics: Foundations of Factory Management[END_REF] as the ratio between the number of production orders with flow times less than or equal to the allotted time (i.e. the DLT), and the total number of production orders (Equation 2); • The customer service rate, defined as the ratio of the number of orders filled on time to the total number of orders; • Flow time distribution, defined as the value where X% of the production orders have flow times less than or equal to this value (where X is 50, 60, 70, 80, 90, and 100); and • The average flow times. These variables will allow us to plot the desired charts presented in the results section. 120 scenarios were simulated for each case. Each scenario is simulated 100 times and each replication lasts 110 weeks (550 days), including a 10-week warm-up period (not included in the results). The modelling and simulation were performed on Arena version 15.10 software.

The industrial case

Workshop parameters for the industrial case

The industrial case is made of two consecutive workshops. In the first one, the raw materials are weighed, then mixed, and heated up in a reactor to manufacture semi-finished products in a tank. In the second one, the semi-finished products are packed on a conditioning line to manufactured finished products, which are shampoo bottles filled up by the semi-finished products (Figure 3).

The raw materials (plant extracts) are buffered and considered available at all times in sufficient quantities. The semi-finished products (shampoos) are buffered, and there are 18 different ones (almond, mint, quinine, etc.). To produce them, the raw materials are weighed at the weighing station, mixed and heated in a reactor, and finally transferred in a tank. There are one weighing station, out of order 10% of the time, and four reactors. The weighing station is the bottleneck resource in this workshop, working 24 h a day (3 shifts of 8 h).

The 95 finished products are buffered. They are made of semi-finished products by being operated on a conditioning line. A semi-finished product can make between 2 and 8 different finished products: The differentiation between finished products from the same semi-finished products comes from the bottle size and the linguistic version. The link between semi-finished products and finished products is shown in Table 4.

The conditioning line is out of order 10% of the time. There are six types of finished products, two according to the bottles' size (200 and 400 mL) and up to 3 different linguistic versions. About half of the finished products of the same semi-finished products is made of 200 mL bottles. The time to change the size of bottles on the conditioning line is about four times the time to change the semi-finished product. Therefore, the decision to change the bottle size is taken the first day of the week only: on Monday, the conditioning line is set to satisfy the longest queue of conditioning orders between those in 200 mL and those in 400 mL.

The DLTs are fixed to 10 days for the first workshop (weighing station and reactors) and 15 days for the second (conditioning line). The LTFs are set to 10% for the semi-finished products buffers and 50% for the finished products buffers. The VF are set to 10% for the semi-finished products buffers and 20% for the finished products buffers. The size of a production order for the semi-finished products are predetermined, depending on the formula and the reactor size (6 or 10 tons). Therefore, we cannot launch orders of 4 tons or 13 tons, for example. The finished products buffers have a MOQ of 5 000 bottles (this is a psychological threshold below which operators and managers do not see the point of launching an order). The workshop manages partial orders the same way as the fictional case.

The second workshop can work 16 h a day (2 shifts of 8 h per week), 20 h a day (1 week with 2 shifts and 1 week with 3 shifts, resulting in an average of 2.5 shifts per week), or 24 h a day (3 shifts). The production manager's goal is to find the best number of shifts to deal with the customer demand. Thus, we will create visual charts to help decide between the three possibilities of the number of shifts (2, 2.5, and 3).

Design of experiments and simulation parameters for the industrial case

For the industrial case, the design of experiments has different goals:

• To compare and validate the fictional case study (the charts must look like the same); • To help the production manager to find the number of shifts with the visual charts, depending on an expected demand; and • To compare the average flow times of the same resource (the conditioning line) between the three capacity possibilities (number of shifts), depending on the resource's loading rate.

To do so, we generate a progressive scale-up to have a load/capacity ratio between 70% and 100% for both workshops and for each capacity possibility for the second workshop. For the chart with the three shifts, we simulate the workshop with an average weekly demand from 75 000 bottles a week to 300 000 bottles a week for each shift.

The outputs variables are the same as the ones for the fictional case: the loading rates of the bottleneck resources, the workshop service rates, the customer service rates (named 'customer service rate' for the second workshop because it serves customers, and 'buffer service rate' for the first workshop because the second one is the customer of the first one), the average flow times and their distributions.

300 scenarios were simulated (60 for the first workshop and 240 for the second one), each of them is simulated 100 times, and each replication lasts 60 weeks, including a 10-week war-up period.

Results and discussion

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For space reasons, only the visual charts are presented.

The ictional lowshop

First, we verify the average flow time of all Production Orders (PO) represents the average flow time of PO for each finished product. Figure 4 illustrates the average flow times of PO for six different finished products (finished products 5, 10, 15, 20, 25, and 30 in different blues) and the average flow time for all PO (in black) depending on three loading rates of the bottleneck resource (70%, 85%, and 95%). It shows an unsubstantial difference between flow times of different finished products. The average flow time all of POs can be used to represent the flow time of POs for all finished products, independently of the loading rate of the resource. We explain it because most of the flow time is flow times (from 50% in light blue to 100% in dark blue); and • On the right y-axis is the service rate scale, between 0 and 100%. The curve represents the workshop service rate at the square marks and the customer service rate by the curve at the circular marks.

From these graphs, it follows that the more the loading rate of the bottleneck resource increases, the more the average flow time increases hyperbolically in the workshop (conclusion known by the queue theory [START_REF] Hopp | Factory Physics: Foundations of Factory Management[END_REF]) but also its dispersion. The average flow time coincides with the area representing 50% of all flow time in the workshop (i.e. the distribution median). Therefore, taking a DLT equal to the average flow time represents only half of the manufacturing orders.

The workshop service rate begins to deteriorate sharply when the DLT line crosses the different flow time distribution areas. The customer service rate is slightly out of line with the workshop service rate due to each stock buffer's red zone, acting as a safety stock.

There is not much difference between the two charts (Figures 5 and6), meaning that machines and operators behave the same way as the bottleneck resource.

These charts help answer the first three research questions. In our example, the DLT set at 10 days can absorb 100% of the flow times up to a load rate of about 91%, 90% of the times up to a rate of 94%. This shows that the average remaining flow time of around 3-5 days should not be relied upon as long as the load rate is below 94%. A 5-day DLT would not even absorb all the flow time for a 70% charge rate.

Then, for example, for a 97% loading rate in Figure 4, the DLT would only absorb 80% of the flow times. As a result, the workshop service rate drops to 80%, and the customer service rate also deteriorates to around 92%. The production manager, anticipating a ramp-up (or ramp-down) on the shop floor, might then choose to adjust the DLT accordingly, or seek to reduce the loading rate by adapting capacity to control flow times (with a target service rate), or increase/decrease safety stocks to absorb flow time variations.

The industrial case

In this case, we verify that the chart looks like the fictional case (to validate our assumptions), and we show how to use the visual charts to help decide on the number of shifts.

Figure 7 has the same axes and legend as the previous charts. It represents the average flow time, dispersion, and service rates depending on the weighing station's loading rate (the bottleneck of the first workshop). The chart is closer to the ones made with the fictional workshop: when the bottleneck resource's loading rate increases, flow times increase too, and service rate decreases. In this case, a DLT set to 10 days covers 100% of the flow times until around 85% of the loading rate. Above this value, the production manager might increase the DLT or seek to reduce the loading rate. To visualise the effect of reducing loading rate, we created a chart for the conditioning line of the second workshop.

In Figure 8, the average weekly customer demand of the workshop is represented in abscissa, and the curves represent the average flow times (full curves), 80% of flow times (dashed curves), and 100% of flow times (dotted curves), when the conditioning line works with 2 shifts (black curves), 2.5 shifts (blue curves) and 3 shifts (green curves). The red dashed line is the Decoupled Lead Time.

As the flow times increases drastically when the demand increases, it seems better to seek to reduce the loading rate (by changing the number of shifts) rather than changing the DLT (what increases the buffer sizing and reduces the shop's reactivity): for example, with a demand of 150 000 bottles a week, a DLT of 15 days absorbs all the flow times with 2.5 shifts while it needs a DLT of 30 days to absorb only the average flow times.

When the average demand increases, the loading rate of the conditioning line increases too, so does the average flow time (and its dispersion). As a result, the production manager could use this chart: if the average demand is not higher than 100 000 bottles a week, 2 shifts and a DLT of 15 days are enough to absorb at least 90% of flow times. If we forecast more than 100 000 bottles a week, the production manager has to increase the DLT, or set 2.5 shifts. Above 200 000 bottles a week, 2.5 shifts are not enough, and we need 3 shifts. Figure 9 represents the average flow times depending on the conditioning line's loading rate for the three sets of shifts.

Firstly, under 98% of loading rate, the three curves are very close, meaning that the average flow times is nearly the same with 2, 2.5, or 3 shifts when the resource is used at the same loading rate (for example, the average flow time is around 5 days at 95% of the loading rate, regardless of the number of shifts).

Secondly, the green curve (3 shifts) turns around when it reaches 98%. The conditioning line set with 3 shifts never get a loading rate above 98%, because the first workshop (weighing station and reactors) becomes the bottleneck workshop. Consequently, as demand increases, conditioning orders are waiting for semifinished products availability, which increases the flowtime while reducing the loading rate of the conditioning stage. To better understand, Figure 10 represents the loading rate of the weighing station (black curve), the weighing workshop buffer service rate (blue curve), and the loading rate of the conditioning line (green curve). Above 250 000 bottles a week, the weighing station loading rate reaches 100%. This is why its service rate begins to decrease (as we saw in Figure 7 previously). Therefore, the semi-finished products are slow to arrive, delaying the production orders of finished products. As a result, the conditioning line is waiting, and its loading rate decreases. This last chart represents the relation between the two workshops, depending on the demand. Above 250 000 bottles a week, the average flow times increase because both the loading rates of the conditioning line are near 98%, and the loading rate of the weighing station reached 100%.

Conclusion

This paper proposes charts to visually correlate bottleneck resources load rate, flow time distribution, and service rates. They are realised by simulating DDMRPmanaged production workshops, including an industrial case. Using the workshop service rate definition as the probability that the flow times are less than or equal to the DLT parameter, we have two choices to improve the workshop service rate: Control the flow times by adjusting the capacity and/or adjust the DLT parameter.

The graphs we have created help deciding based on the expected loading rate or the expected customer weekly demand. It enables mid-term management of capacity as demand trend changes. These experimental results are essentially very contextual. That is why we simulated an industrial case to validate our fictional one.

However, it will be necessary to verify their sensitivity to key workshop parameters: product mix, workshop variability, and technical data. In this study, charts show similar behaviour between the fictional case with bottleneck machine, the fictional case with bottleneck operators, and the industrial case with different numbers of shifts.

The customer service rate is then shifted from the workshop service rate due to the safety stock represented by the DDMRP red zone. Therefore, it is possible to keep a correct customer service rate by playing on this safety stock. However, it requires a strong control of the loading rate. The slightest error induces a drastic drop in the service rate. Consequently, it is more interesting to try to control the flow times rather than to adapt the DLT parameter or to play with the safety stock: a minor capacity adjustment (from 95% to 90%, for example) is equivalent to significantly increasing the DLT (from 25% to 50%) without impacting the average stock. This remark supports the implementation of capacity buffers as suggested by [START_REF] Ptak | Orlicky's Material Requirements Planning 3/E[END_REF], and our charts can be used to determine the size of the capacity buffer.

In our industrial case, the production manager's concern is to find which number of shifts is required to face the demand (the decision is made weekly or monthly). The charts correlate the flow time dispersion depending on the forecasted customer demand and the number of shifts working. They help to decide if 2 shifts are good enough, or if more working hours are needed.

Therefore, it remains up to the production manager to choose which capacity adjustment to implement in the workshop, depending on the available levers (increasing the number of operators, working overtime, working several shifts (switching to 3 shifts for example), capacity subcontracting, etc.). Thus, we can imagine different charts similar to those presented here. Each curve would propose a different scenario (6 operators with 2 shifts or 4 operators with 3 shifts, for example, or even a scenario with an increase in DLT and comparing average in-process and finished product stocks). These graphs would be used as a decision aid for capacity adjustments when forecasting a load increase (or decrease), by setting a customer and/or workshop service rate objective. They could thus be used for tactical or even strategic decisions.

Finally, we showed that the average flow time is not suitable to size the DLT, and it is better to know the dispersion of flow times. By coupling the DDMRP method with a tool reducing the variability of flow times, we could reduce the dispersion, and therefore reduce the DLT, thus reducing stocks while maintaining a high loading rate. Moreover, other types of workshops could also be processed (job shop, open shop, production lines crossovers, etc.) to verify that the same phenomena are identified.

Figure 1 .

 1 Figure 1. The three zones of a DDMRP stock buffer.

Figure 2 .

 2 Figure 2. Diagram of the studied production line and positioning of the DDMRP stock buffers.

Figure 3 .

 3 Figure 3. Diagram of the studied industrial case composed of two workshops and positioning of the DDMRP stock buffers.

Figure 4 .

 4 Figure 4. Average Flow Times of Production Orders for different Finished Products depending on the Loading Rate of the Bottleneck Resource.

Figure 5 .

 5 Figure 5. Flow Times (average and distribution) and Service Rates (workshop and customer) depending on the loading rate of machine 5.

Figure 6 .

 6 Figure 6. Flow Times (average and distribution) and Service Rates (workshop and customer) depending on the loading rate of the operators.

Figure 7 .

 7 Figure 7. Flow Times (average and distribution) and Service Rates (workshop and buffer) depending on the loading rate of the weighing station.

Figure 8 .

 8 Figure 8. Flow Times (average, 80%, and 100%) depending on the Average Weekly Demand (in bottles) and the number of shifts (2, 2.5, 3).

Figure 9 .

 9 Figure 9. Average Flow Time depending on the Loading Rate of the Conditioning Line and the number of shifts (2, 2.5, 3).

Figure 10 .

 10 Figure 10. Loading Rates of the Weighing Station and the Conditioning Line, and Service Rate of the Weighing Workshop Buffer depending on the Average Weekly Demand (in bottles).

Table 1 .

 1 Changeover time (in hours) for each product.

Table 2 .

 2 Production time (in seconds per part) for each product.

	Products	M1	M2	M3	M4	M5	M6
	1-10	5	5	5	10	15	5
	1 1 -2 0	5	1 0	1 0	1 0	1 0	1 0
	21-30	5	15	15	10	5	15

fifth machine to 6-hour outages every 34 h, representing 10% and 15% of the time respectively. These values arbitrarily follow exponential laws to model a non-negligible but not excessive variability in production, close to reality.

Table 3 .

 3 Average order size for each product.

	P r o d u c t s	1	2	3	4	5
	A D U	6 5	2 3	5 5	6 3	1 9
	Products	6	7	8	9	10
	A D U	3 8	2 9	3 3	6 1	3 4
	Products	11	12	13	14	15
	A D U	6 1	6 2	5 7	3 2	5 2
	Products	16	17	18	19	20
	A D U	2 1	6 1	4 2	4 8	4 4
	Products	21	22	23	24	25
	A D U	3 5	5 2	2 3	5 6	5 5
	Products	26	27	28	29	30
	A D U	3 7	3 2	5 1	5 5	5 2

Table 4 .

 4 Links between semi-finished products and finished products.

	Semi-finished		Finished	Semi-finished		Finished
	product	Bottle size	products	product	Bottle size	products
	1	200	1-3	10	200	50-52
		400	4-5		400	53-55
	2	200	6-8	11	200	56-58
		400	9-11		400	59-60
	3	200	12-14	12	200	61-62
		400	15-17		400	63-64
	4	200	18-20	13	200	65-67
					400	68-70
	5	200	21-23	14	200	71-73
		400	24-26		400	74-75
	6	200	27-29	15	200	76-77
		400	30-31		400	78-79
	7	200	32-34	16	200	80-82
		400	35-37		400	83-85
	8	200	38-40	17	200	86-88
		400	41-43		400	89-93
	9	200	44-46	18	200	94-95
		400	47-49

Data availability statement

The data that support the findings of this study are available from the corresponding author, Guillaume Dessevre, upon reasonable request. Restrictions apply to the availability of these data, which are available with the permission of the industrial partner.

Disclosure statement

No potential conflict of interest was reported by the author(s).


Walid Al-Maksoud

Stéphane Menuel

Mohamad Jahjah

Eric Monflier

Catherine Pinel

Laurent Djakovitch
email: laurent.djakovitch@ircelyon.univ-lyon1.fr

Base directed palladium catalysed Heck arylation of acrolein diethyl acetal in water

The selective Heck arylation of acrolein diethyl acetal catalysed by [Pd(NH 3 ) 4 ]Cl 2 in the presence of RAME-␤-CD in water as solvent is described. Depending on the base (i.e. NaOAc or HN(i-Pr) 2 ) good to high selectivity's towards, respectively, the cinnamaldehydes 2 or the 3-arylpropionic esters 1 were achieved. The results support that depending on the base different palladium intermediate complexes are formed. Using NaOAc, {[ArPdX(H 2 O) 2 ]} complex is preferentially generated giving the cinnamaldehyde 2. On the other hand, in the presence of HN(i-Pr) 2 , a L-type ligand, [ArPdX(HN(i-Pr) 2 (H 2 O)] or [ArPdX(HN(i-Pr) 2 (HN(i-Pr) 2 )] will be generated leading to the formation of the 3-arylpropionic ester 1. For the last, coordinated amine participates very probably to the formation of the esters through intramolecular syn ␤-H elimination.

Introduction

Cinnamaldehyde derivatives found applications in food, cosmetic agrochemical and pharmaceutical industries while 3-aryl propionic acids are common building blocks in organic synthesis. Therefore, several multi-step and costly organic methods have been reported for their synthesis [START_REF] Rosen | Comprehensive Organic Synthesis[END_REF][START_REF] Garbe | Ullmann's Encyclopedia of Industrial Chemistry[END_REF]. These have been supplanted by palladium-catalysed Heck procedures [START_REF] Djakovitch | [END_REF], mainly devoted to cinnamaldehydes, from allylic alcohol [4][5][6][7] or acrolein [8][9][10]. In the last, some degradation was observed [11] that can be avoided by using acrolein dialkyl acetal [8]. However, the reaction was not selective and mixtures of aldehyde and ester were obtained those selectivity could be directed by the nature of the catalytic system. For example, the group of Cacchi reported a procedure in DMF at 90 • C to prepare selectively cinnamaldehydes (Pd(OAc) 2 , Bu 4 NOAc, K 2 CO 3 , KCl) [12] or 3-arylpropionic ester (Pd(OAc) 2 , n-Bu 4 NCl, n-Bu 3 N) [13]. Similarly, Santelli and Doucet reported the synthesis of cinnamaldehydes or 3-arylpropionic from 3,3-diacetoxypropene [14] or acrolein ethylene acetal [15].

In previous works, we extended this approach to a large variety of condensed aryl and heteroaryl substrates using either homogeneous [10,16] or heterogeneous [17] palladium catalysts. Under similar conditions to ours, Najera and Botella reported the preparation of cinnamaldehyde derivatives using a dimeric 4-hydroxyacetophenone oxime-derived palladacycle as catalyst [18,19]. While successful, these palladium catalysed methodologies remain linked to the use of polar organic solvents like DMF or DMAc that have been recently prohibited in most of the chemical industries.

In this context, the use of alternative non-toxic solvents appears to be very attractive. Water emerges as a good candidate in regards to previous industrial development [START_REF] Cornils | Aqueous-Phase Organometallic Catalysis[END_REF] and successful transposition of Heck arylation of olefins in this media [START_REF] Lamblin | [END_REF]22]. Initially, Beletskaya and co-workers reported the arylation of acrylic acid by diaryliodonium salts in presence of Pd(OAc) 2 and Na 2 CO 3 in water at 100 • C leading to significant yields (50-94%) of cinnamic acids [23]. The methodology was then adapted to styrene arylation [24,25] and was regarded as a green approach for the preparation of cinnamic derivatives [26]. Extension of these pioneering work concerned mainly development of water soluble systems based on hydrophilic phosphine [27] or reaction under microwave irradiations using hypervalent iodonium salts [28].

These examples, while interesting, remain quite limited due to the exclusive use of water soluble substrates to achieve good yields. Some authors used additives in order to circumvent such limitation in which the use of phase transfer agent like tetraalkyl ammonium salts [29,30] appeared particularly successful. Originally, Jeffery reported the coupling of iodobenzene with acrylates in the presence of tetrabutyl ammonium salts [29], followed by Tsai and co-workers who used palladium cationic complexes in the presence of TBAB [30]. Recently, Iranpoor and al. reported the use of phosphazane ligands for base-free Pd(II) catalysed Heck coupling reaction of aryl iodides, bromides and chlorides in water with styrene, n-butylacrylate, 1-octene and cyclohexene. The palladium complex is easily separated by filtration and reused for several runs [31].

Pursuing our efforts to develop sustainable processes for fine chemical syntheses we investigated in detail the Heck arylation of acrolein diethyl acetal to cinnamaldehydes or 3-arylpropionic acids in water (Scheme 1) [32][33][34][35]. The effect of various cyclodextrins on reaction rate and selectivity was also evaluated. Indeed it is known that cyclodextrins can play a positive role in large range of water phase reaction [32][33][34][35], to the best of our knowledge no study was reported on the Heck arylation in the presence of homogeneous catalysts despite the few existing examples concerning the use of heterogeneous catalyst [36][37][38][39].

Experimental

They include the general information, the preparation of the catalysts, the syntheses and the analytical data ( 1 H NMR, 13 C NMR, Mass Spectra) relative to the cyclodextrins and the procedures for the catalytic tests. Mars microwave of CEM Corporation was used for the syntheses of cyclodextrin. It delivers an energy output of 800 w at a frequency of 2450 MHz. Teflon microwave-transparent XP-1500 vessels with inboard RTP-300 Plus temperature sensor control were used. With this configuration, the system regulates the microwave power out-put to maintain the desired reaction temperature.

2.1. Procedure for the synthesis of ˇ-CD-NEt 2 ␤-CD-OTs (2.0 g, 1.56 mmol) in diethylamine (60 mL) was stirred and irradiated at 90 • C in closed reactor for 2 h under microwaves. After total evaporation of the amine, the crude product was dissolved in minimum distilled water and heated at 60 • C Cooling the solution to 20 • C resulted in precipitation of white crystalline needles. Filtration on a glass filter and subsequent drying under vacuum gave ␤-CD-NEt 2 with 76% yield.

Procedure for the synthesis of RAME-ˇ-CD-OTs

NaOH solution (18 g, 450 mmol, in 50 mL of water) was added dropwise to a suspension of ␤-CD-OTs (19 g, 15 mmol) in water (50 mL) at 0 • C over a period of 10 min. A solution of dimethylsulfate (90 mL, 950 mmol) in THF (15 mL) at 0 • C was then added dropwise to the resulting clear solution. Ethanol (10 mL) was added to the solution which was brought to room temperature and stirred for 5 h. After, THF was evaporated and the product was extracted from the aqueous phase with chloroform (3 × 500 mL). The organic phase was dried with anhydrous MgSO 4 . Chloroform was removed under reduced pressure to give RAME-␤-CD-OTs with 57% yield.

Procedure for the synthesis of RAME-ˇ-CD-NEt 2

RAME-␤-CD-OTs (2.5 g, 1.56 mmol) in diethylamine (60 mL) was stirred and irradiated at 90 • C on closed reactor for 2 h under microwaves. After total evaporation of the amine, the crude product was dissolved in 50 mL of distilled water and the product was extracted from the aqueous phase with chloroform (3 × 250 mL). The organic phase was dried with anhydrous MgSO 4 . Chloroform was removed under reduced pressure to give RAME-␤-CD-NEt 2 with 52% yield.

Procedure for the catalytic tests

1 mmol of aryl halide, 3 mmol of acrolein diethyl acetal, 1.5 mmol base and 2 mol% of Pd-catalyst, and when applied xmol% CDs were introduced in a pressure tube. 2 mL water was added and the reactor was then placed in a pre-heated oil bath at 100 • C for 24 h under vigorous stirring. After cooling to room temperature few drops HCl 1 N were added to the reaction mixture that was extracted with 10 mL of a standard solution as EtOAc/dodecane (0.001 M). The organic layers were dried over MgSO 4 (an aliquot is removed to perform GC analyses) and evaporated under reduced pressure and the residue was then purified by flash chromatography on silica gel.

Procedure for the catalysts recycling tests

In a typical experiment fresh catalyst was used as for a standard catalytic run (2 mmol aryl halide, 6 mmol acrolein diethyl acetal, 3.5 mmol base, 2 mol% Pd-catalyst, 4 ml water, 90 • C, 24 h). After 24 hours reaction an aliquot was removed from the reaction mixture in order to perform GC-analyses and new amounts of reagents (2 mmol of aryl halide, 6 mmol of acrolein diethyl acetal, 3.5 mmol base) were added. The volume of solvent was adjusted to initial volume in order to restore the concentrations of reagents to that of the initial run. Immediately after addition, based on GC analyses, the concentration of the aryl halide was set to 100% and the concentration in products to 0%. The reaction was followed by GC for another 24 h and the procedure was repeated four times.

Results and discussion

Initial studies

Several commercially available palladium salts (i.e. Pd(OAc) 2 , PdCl 2 , Na 2 PdCl 4 , Pd(NH 3 ) 4 Cl 2 ) and home-made palladium catalyst {Pd[P(o-Tol) 3 (OAc)]} 2 (i.e. Herrmann-Beller palladacycle; prepared following reported procedure [40]) were evaluated and compared one with each other's in the Heck arylation of acrolein diethyl acetal in pure water.

In some reactions, cyclodextrins were used as additive in order to increase the catalyst activity. For this purpose, several structurally different cyclodextrins (Fig. 1) were evaluated. Except the Scheme 1. commercially available ␤-cyclodextrin (i.e. ␤-CD), RAME-␤-CD and 2-hydroxypropyl-␤-cyclodextrin (i.e. HP-␤-CD), ␤-CD-NH 2 [41], ␤-CD-trz-CH 2 OH [42], RAME-␣-CD [43] and RAME-␥-CD [43] were synthesised and characterised according procedures previously described.

The synthesis of ␤-CD-NEt 2 has been carried out in one step from the ␤-CD-OTs derivative under microwaves (MW) irradiation and easily isolated after recrystallization from hot water in 76% yield.

The RAME-␤-CD-NEt 2 has been obtained in two steps from the ␤-CD-OTs. In the first step, the RAME-␤-CD-OTs was synthesised by reaction of ␤-CD-OTs in alkaline conditions at 0 • C with a dimethyl sulfate solution. After work-up, RAME-␤-CD-OTs was isolated as a pale yellowish-white powder in 57% yield. For the second step, RAME-␤-CD-OTs was dissolved in diethylamine and the solution was heated up to 90 • C in a closed reactor using microwave irradiation for 2 h. The randomly RAME-␤-CD-NEt 2 was obtained in 52% yield as a mixture of partially methylated ␤-CDs with an average degree of substitution of 1.8 methyl groups per glucopyranose unit.

In order to discern which reaction conditions allowed to perform the Heck arylation of acrolein diethyl acetal in water, we realised a preliminary study comparing the performances of various sources of palladium in two solvents, namely NMP previously described [17] and water. Iodobenzene was chosen for this set of reaction (Table 1). Based on our previous results [17], the reaction was carried out for 24 h using sodium acetate as base.

As expected, all reaction carried out in NMP led to high conversions and good to high selectivities (i.e. >65%) towards the 3-arylpropionic ester 1 (Table 1, entries 1, 3) independently on the source of the palladium catalyst. In water, the results are quite different. Generally moderate conversion was achieved using the Herrmann-Beller palladacycle (i.e. 40%); Table 1, entries 2) whereas low conversion (i.e. <15%; Table 1, entries 4) was observed when engaging the Pd(OAc) 2 as catalyst. Noticeably, in water reverse selectivity towards aldehyde 2 is observed (i.e. selectivity >80%). Whatever the Pd-catalyst used, the conversion remained low (i.e. <40%). In a few cases (Table 1, entries 5, 6 and 7) saturated aldehyde 3 was also observed in low amount, mainly when palladium chloride derivatives were engaged. Presumably this compound is formed according to mechanism depicted in Fig. 2 [44]. After formation of the C C bond, the palladium complex A isomerises to the enolate intermediate B that yields the observed saturated aldehyde 3 after hydrolysis.

Even if modest conversions were observed, these preliminary results have shown that Heck coupling of acrolein diethyl acetal in water is feasible.

Adding cyclodextrins

Among all evaluated catalysts, the [Pd(NH 3 ) 4 ]Cl 2 catalyst was chosen to evaluate further various reactions parameters for converting acrolein diethyl acetal and iodobenzene to cinnamaldehydes 2 in water in the presence of NaOAc. Here, we were interested in improving the efficiency of the catalytic system to propose a procedure competitive to those using organic solvent.

With this aim, we evaluated the influence of various cyclodextrins (Fig. 1) on both the conversion and the selectivity of this transformation (Table 2).

The nature of cyclodextrins has a dramatic influence on the evolution of the reaction. While addition of RAME-␣-CD and RAME-␥-CD induced lower conversion (22% and 24%, respectively; Table 2, entries 9-10); addition of RAME-␤-CD allowed increasing noticeably the conversion of iodobenzene (86% vs 33%; Table 2, entries 3 and 1, respectively). Regardless of the cyclodextrin, the selectivity of the reaction remained unchanged (75-78% in favour of aldehyde 2). Considering the positive effect of ␤-cyclodextrin family, we evaluated a range of substituted ␤-CDs on the conversion and the selectivity of the reaction. ␤-CD-NH 2 and ␤-CD-trz-CH 2 OH increased slightly the conversion up to 39% (Table 4, entry 7) and 52% (Table 2, entry 4), respectively, while native ␤-CD allowed increasing the conversion by a factor 2 (Table 2, entry 2). In a similar fashion, HP-␤-CD, ␤-CD-NEt 2 and RAME-␤-CD-NEt 2 promoted efficiently the reaction achieving high conversions (96-100%), with, however, a strongly decreased selectivity when using RAME-␤-CD-NEt 2 . To summarise, the best result was obtained with the randomly easily available methylated RAME-␤-CD that gave 86% conversion without affecting the selectivity (78% in 2) (Table 2, entry 2).

Next we evaluated the influence of the RAME-␤-CD loading (i.e. the CD/iodobenzene ratio) on the conversion and the selectivity under previously optimised reaction conditions (Fig. 3). The selectivity of the reaction is constant during the course of the reaction and is not affected by adding cyclodextrin as it remains relatively close in all reaction sets in favour of the cinnamaldehyde 2. The small differences observed are in the error range of the analytical method (72-78%).

Influence of the nature of the base

We decided to investigate deeply the influence of the nature of the base (i.e. organic and inorganic bases) on the Heck arylation of acrolein diethyl acetal with iodobenzene, firstly without additive. The use of mineral bases like hydroxide, carbonate and phosphate salts did not allow enhancing the conversion compared to NaOAc, except for NaOH that gave 61% of conversion (Table 3, entries 6-9). In all cases, aldehyde 2 is mainly formed (>90%, entries 5-8). Tertiary amines like Et 3 N or EtN(i-Pr) 2 led generally to moderate to good conversions (i.e. 80%); however with a mixture of aldehyde 2 and ester 1 (Table 3, entries 2 and 5).

Unexpectedly, the use of diisopropylamine resulted in high activity together with a complete reverse selectivity in favour of the 3-arylpropionic ester 1. Another issue concerned the conversion of iodobenzene that increased considerably reaching quantitative conversion after 5 h (Table 3, entry 3). In that case it was demonstrated that reducing the Pd-catalyst loading down to 0.5 mol% [Pd(NH 3 ) 4 ]Cl 2 (in regards to iodobenzene) did not strongly affect the efficiency of the system since 83% of conversion was achieved within 24 h (Table 3, entry 4). Very few reports concern the use of such secondary amines in Heck reaction. Dawood and al. reported the use of NH(i-Pr) 2 in Heck coupling of 4-bromoacetophenone with tButyl acrylate in DMF and water. In water, they observed a full conversion using NH(i-Pr) 2 whereas only 5% was obtained with NaOAc [45]. Similar trends were observed in our hands.

To sum up, in water using NaOAc as the base gave mainly aldehyde 2 in moderate conversion while diisopropylamine led to the formation of ester 1 in high yields. Thus, the nature of the base influenced not only the activity of the reaction but also its selectivity. Base on literature reports and according to the experimental observations, it is rather reasonable to propose that the reverse selectivity results from the respective electronic nature of intermediate palladium complexes generated under the reaction conditions. In organic solvent, the formation of ester takes place via the syn-␤-hydride elimination of hydrogen borne by the acetal while that of aldehydes results from syn-␤-hydride elimination of benzylic hydrogen [17,46]. In water, the situation is quite different. Whatever the initial "naked" Pd-species generated at the first stage of the reaction, after oxidative addition solvent participate to the stabilisation of the ArPd (II) X complex through coordination. In the absence of other L-type ligand, {[ArPdX(H 2 O) 2 ]} complex will be preferentially generated (Fig. 4, route (a)). This complex will led, after olefin coordination and insertion, to the formation of aldehyde 2 since internal Pd-coordination by the ethoxy group in intermediate C would prevent the internal rotation along PdCH-CH(OEt) 2 axis.

On the other hand, in the presence of diisopropylamine as base, according to Jutand's reports [47], [ArPdX(HN(i-Pr) 2 (H 2 O)] (Fig. 4, route (b)) or [ArPdX(HN(i-Pr) 2 (HN(i-Pr) 2 )] (Fig. 4, route (c)) will be generated as NH(i-Pr) 2 is a stronger coordinating ligand than water, and given the Pd/NH(i-Pr) 2 ratio (1/175). In both cases, amine coordination to Pd will prevent further internal coordination by ethoxy groups due to steric hindrance and would thus favour the internal rotation along the PdCH-CH(OEt) 2 axis. This last step could be accelerated by intramolecular hydrogen bond (NH• • •OEt) between the diisopropylamine ligand and the ethoxy group. Additionally, it is strongly suggested that coordinated amine participates intramoleculary to the syn ␤-hydrogen elimination via the H gem to the diacetal yielding thus the ester. With these results in hand, likely to NaOAc, we evaluated the influence of adding cyclodextrin when using NH(i-Pr) 2 as the base (Table 4).

In the presence of NH(i-Pr) 2 the nature of CD impacted the performances of the catalytic system: native ␤-CD led to decrease activity while RAME-␤-CD resulted in slightly increased efficiency (Table 4, entries 1-3). These observations were confirmed by kinetic investigations performed at 90 • C: while the highest activity of the reaction performed in absence of CDs arises 0.45 mol/mol [Pd] /min that in the presence of native ␤-CD drops to 0.25 mol/mol [Pd] /min. On the contrary, the use of RAME-␤-CD did not affect the efficiency of the catalytic system (A max = 0.45 mol/mol [Pd] /min). Interestingly, addition of RAME-␤-CD allowed stabilising the catalytic system as quantitative conversion of iodobenzene was observed after 3 h versus 5 h in its absence (see Fig. S1 in supporting information). Decreasing the reaction temperature allowed to observe further the positive influence of adding RAME-␤-CD as clearly higher conversion was achieved versus the time in presence of RAME-␤-CD (Table 4, entries 7 versus 7).

The selectivity of the reaction in favour of the ethyl 3phenylpropionate 1 was not affected by adding cyclodextrins to the reaction medium.

Encouraged by this stabilising effect, we evaluated the recycling of the reaction medium using the following procedure: after a first run with the fresh catalytic system (24 h), new amounts of reagents (i.e. 2 mmol PhI, 6 mmol acrolein diethyl acetal and 3.5 mmol NH(i-Pr) 2 ) were added to the reaction mixture. At the same time, the solvent volume was restored to initial volume in order to recover the same reagent and palladium concentrations as in the initial run (see Experimental for detailed procedure). Fig. 5 clearly indicated that addition of RAME-␤-CD allowed higher recycling potential, demonstrating the stabilising effect. Unfortunately, even in the presence of cyclodextrin, some deactivation, that could not be fully related to mass loss due to sampling, was observed but to a lesser extent. It could be attributed either to effective catalyst deactivation due to the formation of low active palladium aggregates or to the higher viscosity of the solution that prevent efficient mass transfer between the aqueous and organic phase.

To end, with NH(i-Pr) 2 as the base, we examined the influence of the RAME-␤-CD/iodobenzene ratio when decreasing the Pdcatalyst loading to 0.5 mol% (Table 5). A stark effect was observed as generally 6-7 h versus 24 h were enough to achieve quantitative conversion of iodobenzene. Thus, the stabilising effect provided by using RAME-␤-CD allowed reducing the palladium concentration without affecting neither the conversion nor the selectivity.

Influence of the nature of the aryl halide

Finally we evaluated the influence of electron-donating or withdrawing substituents on the aromatic ring (Table 6).

To summarise the results with iodobenzene, using NH(i-Pr) 2 as the base high conversions and high yields in ester 1 were obtained (Table 6, entries 3-4) whereas in presence of NaOAc as the base the selectivity turn to the aldehyde 2; however in low conversion except when RAME-␤-CD was used as additive (Table 6, entries 1-2). To extend further applications of this procedure, para-iodoanisole was engaged as a representative of the electron donating derivatives. In the presence NH(i-Pr) 2 as the base, high conversion towards the expected corresponding ester 1 was achieved; adding RAME-␤-CD in the reaction mixture allowed to reduce the reaction time from 24 h to 8 h (Table 6, entries 7-8). With this substrate, we performed a set of reaction using NaOAc as the base. As could be expected, lower conversions were achieved despite the use of RAME-␤-CD; but high selectivity towards the aldehyde 2 was observed (Table 6, entries 5-6).

With arylbromides the results are strongly linked to the nature of the substituents, according to Tsai and co-workers who used TBAB as additive for the Heck arylation of n-butyl acrylate in water [30]. Bromobenzene was found unreactive under all applied reaction conditions leading to traces of products (Table 6, entries 9-12). Particularly, using NaOAc as the base very low conversions were observed, a result that was found to be quite general with substituted aryl bromides, encouraging us to evaluate further only the reaction sets with NH(i-Pr) 2 as the base.

Thus, whatever the nature of the aryl halide, high conversions and good (i.e. 60%) to high (i.e. 92%) selectivities towards the ester 1 were achieved when NH(i-Pr) 2 was used as the base. As expected, electron withdrawing substituents such as NO 2 (Table 6, entries [13][14][15][16] or COCH 3 (Table 6, entries [START_REF] Lamblin | [END_REF][22] gave the highest conversion under described reaction conditions. Only for ortho substituents, higher reaction temperature (i.e. 120 • C) allowed us to achieve full conversion in 24 h (Table 6, entries [13][14]. Generally, for these derivatives, the use of RAME-␤-CD allowed to reduce the reaction time from 24 h to 6 h for complete conversion. Only the case of CF 3 substituted derivatives gave unexpected low conversions (40% for the meta-derivative and 50% for the para-derivative -Table 6 entries 17 and 19) despite the high electron withdrawing property of this substituent. This can be noticeably improved by adding RAME-␤-CD in the reaction mixture (i.e. 70% and 66%, respectively Table 6 entries 18 and20).

Conclusion

At this stage of our studies, it is interesting to compare the in and ex-aqua methodologies for the Heck arylation of acrolein diethyl acetal. Both media allowed controlling the selectivity of the reaction towards either the cinnamaldehydes or 3-arylpropanoates either by adapting the catalytic system in organic solvent or by changing the nature of the base when working in water.

To conclude, a procedure for the arylation of acrolein diethyl acetal by the Heck reaction catalysed by the ligandless commercially available [Pd(NH 3 ) 4 Cl 2 ] in pure water have been developed. Remarkably, by just changing the base we were able to achieve, after optimisation, either the synthesis of cinnamaldehydes 2 {[Pd], NaOAc, RAME-␤-CD} or that of ethyl 3-arylpropionate 1 {[Pd], NH(i-Pr) 2 }.

The study revealed that the use of CD was critical in the synthesis of cinnamaldehydes to achieve high conversions of aryl halides with high selectivity towards the expected compound. Unfortunately, this study pointed also out that it was limited to aryl iodides since aryl bromides were found to exhibit lower reactivity.

Fig. 1 .

 1 Fig. 1. Cyclodextrins used in the study. DS: degrees of substitution representing the average number of substituants (CH3 or CH2CHOHCH3 groups) per glucopyranose unit on the C-2, C-3 and C-6 positions.

Table 1

 1 Influence of the palladium catalyst on the Heck arylation of acrolein diethyl acetal with iodobenzene. PhI, 3 mmol acrolein diethyl acetal, 1.5 mmol NaOAc, 2mol% [Pd], 2 ml solvent, reflux, 24 h. a Conversion and selectivity determined by GC. Dodecane was used as external standard for GC-quantitative determination. b Ar-Ar < 10%.

Fig. 2 .

 2 Fig.2. Proposed pathways to account for the formation of 3 (based on reference[44]).

Fig. 3

 3 Fig. 3 clearly indicates the positive influence of adding RAME-␤-CD. Since addition of 1 mol% CD, the initial activity of the reaction increases by a factor 2 to rise to 0.14 mol/mol [Pd] /min. Increasing the CD/iodobenzene ratio resulted in enhanced activity and with 20 mol% RAME-␤-CD a gap in activity that arises 0.28 mol/mol [Pd] /min was observed.The selectivity of the reaction is constant during the course of the reaction and is not affected by adding cyclodextrin as it remains relatively close in all reaction sets in favour of the cinnamaldehyde

Fig. 3 .

 3 Fig. 3. Influence of RAME-␤-CD loading on the conversion of iodobenzene versus the time. Selectivities (product: 1/2/3; see scheme in Table 1) are indicated on the curves; Initial activities depending on RAME-␤-CD loading are given on histograms. Reaction conditions: 2 mmol PhI, 6 mmol acrolein diethyl acetal, 3 mmol NaOAc, 2 mol% [Pd(NH3)4Cl2], 0-20 mol% CD, 4 ml water, T = 100 • C.

Fig. 4 .

 4 Fig. 4. Proposed mechanisms to account for the formation of either the aldehydes 2 or the 3-arylpropionic ester 1 in the Heck arylation of acrolein diethyl acetal in water depending on the nature of the base.

Fig. 5 .

 5 Fig. 5. Influence of RAME-␤-CD on recycling. Reaction conditions: run 1: 2 mmol PhI, 6 mmol acrolein diethyl acetal, 3.5 mmol NH(i-Pr)2, 2 mol% [Pd(NH3)4Cl2], 4 ml water, 90 • C; run 2-4: adding 2 mmol PhI, 6 mmol acrolein diethyl acetal, 3.5 mmol NH(i-Pr)2 and adjusting the solvent volume to the initial value.

Table 2

 2 Influence of cyclodextrins on the Heck arylation of acrolein diethyl acetal with iodobenzene.

	.

a Conversion and selectivity determined by GC. Dodecane was used as external standard for GC-quantitative determination. b Ar-Ar < 10%.

Table 3

 3 Influence of base on the Heck arylation of acrolein diethyl acetal with iodobenzene in pure water catalysed by the [Pd(NH3)4]Cl2.

	.

Reaction conditions: 1 mmol PhI, 3 mmol acrolein diethyl acetal, 1.5 mmol Base, 2 mol% [Pd], 2 ml water, reflux, 24 h.

a Conversion and selectivity determined by GC. Dodecane was used as external standard for GC-quantitative determination. b 0.5 mol% [Pd].

Table 4

 4 Influence of cyclodextrins on the Heck arylation of acrolein diethyl acetal with iodobenzene. Conversion and selectivity determined by Dodecane was used as external standard for GC-quantitative determination.

	.

a

Table 5

 5 Influence of RAME-␤-CD loading on the Heck arylation of acrolein diethyl acetal with iodobenzene.

	.

a Conversion and selectivity determined by GC. Dodecane was used as external standard for GC-quantitative determination.

Table 6

 6 Heck arylation of acrolein diethyl acetal with substituted aryl halides in water.Reaction conditions: 2 mmol ArX, 6 mmol acrolein diethyl acetal, 3.5 mmol base, 2 mol% [Pd](NH3)4Cl2, 0 mol% or 10 mol% RAME-␤-CD as specified, 4 ml water, reflux, 24 h.a Conversion and selectivity determined by GC. Dodecane was used as external standard for GC-quantitative determination.

	.

b 8 h. c 120 • C d 6 h.

Acknowledgement

The authors gratefully acknowledge the National Agency of Research (CATAQ N • ANR-07-CP2D-0167-01/02) for funding.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.apcata.2013.10.004.