Daniele Astolfi

Marc Jungers

Luca Zaccarian

Output Injection Filtering Redesign in High-Gain Observers

Keywords:

We propose a new paradigm to redesign high-gain observers in order to improve performances in the presence of measurement noise. In particular, instead of driving the observer by means of a standard output injection term, we filter it with a dynamical system having good filtering properties. In this first preliminary result we also select the filter in order to address numerical challenges.

I. INTRODUCTION

Among different techniques for the state-estimation of continuous-time nonlinear systems, high-gain observers are certainly very popular in nonlinear control theory. After seminal works at the end of the 80's, a large number of researchers devoted their attention to the study of this class of observers, see, for instance, [START_REF] Gauthier | Deterministic observation theory and applications[END_REF]- [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF] and references therein. The main advantage of a high-gain observer is that it guarantees that the estimation error converges exponentially to zero with a rate of convergence that can be arbitrarily augmented by enlarging just one single parameter, denoted as highgain parameter. However, because of the fast convergence, sensitivity to high-frequency measurement noise is made worst, see [START_REF] Vasiljevic | Differentiation with high-gain observers the presence of measurement noise[END_REF]- [START_REF] Astolfi | Sensitivity to High-Frequency Measurement Noise of Nonlinear High-Gain Observers[END_REF]. To address this problem, many techniques have been developed in the last decades: adaptive techniques, [START_REF] Ahrens | High-gain observer in the presence of measurement noise: a switched-gain approach[END_REF], [START_REF] Sanfelice | On the performance of high-gain observers with gain adaptation under measurement noise[END_REF], redesign of the local behaviour by combining different observers, [START_REF] Boizot | An adaptive high-gain observer for nonlinear systems[END_REF]- [START_REF] Astolfi | Uniting local and global observers for the state estimation of nonlinear continuous-time systems[END_REF], low-power structures, [START_REF] Astolfi | A High-Gain Nonlinear Observer with Limited Gain Power[END_REF]- [START_REF] Khalil | Cascade high-gain observers in output feedback control[END_REF], and lastly saturation redesign, [START_REF] Astolfi | Stubborn ISS Redesign for Nonlinear High-Gain Observers[END_REF]. In the recent work [START_REF] Khalil | Analysis of the Use of Low-Pass Filters with High-Gain Observers[END_REF], it is proposed to filter the measured output by means of a fast low-pass filter, that is a low-pass filter which is modulated by the high-gain parameter. The main drawback of this approach, however, is that asymptotic (possibly practical in presence of measurement noise) estimation cannot be any more guaranteed and the proposed scheme guarantees good performances only with feedback stabilization. Therefore, in the case where we are interested in purely estimation, that scheme can not be used any longer.

In this work, inspired by [START_REF] Astolfi | A High-Gain Nonlinear Observer with Limited Gain Power[END_REF] and in the same spirit [START_REF] Khalil | Analysis of the Use of Low-Pass Filters with High-Gain Observers[END_REF], we propose to use a low-pass filter in order to improve performances of high-gain observers in the presence of measurement noise while retaining the main behaviours. In the proposed design, the dynamical filter processes the output injection term of the observer and not the measured output, as in [START_REF] Khalil | Analysis of the Use of Low-Pass Filters with High-Gain Observers[END_REF]. Sufficient conditions for design of the matrix coefficients, ensuring convergence of the observer, are 1 Université de Lorraine, CNRS, CRAN, F-54000 Nancy, France {daniele.astolfi@univ-lorraine.fr, marc.jungers@univ-lorraine.fr}.

2 LAAS-CNRS, Université de Toulouse, CNRS, Toulouse, France, and Dep. of Industrial Engineering, University of Trento, Trento, Italy {zaccarian@laas.fr}.

This project has received funding from the "Région Grand-Est" and from CaRiTRo, grant PowerLyap.

provided. The estimation error provided by the new structure retains the same input-to-state properties of standard highgain observers. A special choice of this dynamical filter is also given, where we exploit the possibility of writing all the coefficients embedding powers of the high-gain parameter of order 1 and 2 instead of n, similarly to the low-power structures presented in [START_REF] Astolfi | A High-Gain Nonlinear Observer with Limited Gain Power[END_REF]- [START_REF] Khalil | Cascade high-gain observers in output feedback control[END_REF]. In this case, the choice of the matrix coefficients must satisfy a necessary condition, which is fully exploited. With respect to [START_REF] Astolfi | A High-Gain Nonlinear Observer with Limited Gain Power[END_REF]- [START_REF] Astolfi | Peaking-free low-power high-gain observers[END_REF], the procedure of selection of the coefficients in the proposed structure is much simpler. Simulations are enclosed at the end of the work.

We highlight that the goal of this preliminary work is to present a new structure for the design of high-gain observer that aim to address the problem of state observation in noisy frameworks. For this reason, we are not going to compare the proposed techniques with other existing high-gain observer techniques (such as [START_REF] Astolfi | A High-Gain Nonlinear Observer with Limited Gain Power[END_REF], [START_REF] Khalil | Analysis of the Use of Low-Pass Filters with High-Gain Observers[END_REF]) but we will show that dynamical filtering may be effective in improving disturbance rejection properties while inheriting the asymptotic behaviours of the observers. Optimalitybased tuning for both the gain and the parameters of the dynamical filter remains an open question for future research.

Notation. We denote with R the set of real numbers and with Z the set of integers. We denote also Z >0 = {1, 2, . . .}. We denote with C the set of complex numbers and C <0 the set of complex numbers with strictly negative real part. We denote with i the imaginary number. Given a complex number z ∈ C, z = x + iy, with x, y ∈ R, we denote with (z) its real part, namely (z) = x and (z) its imaginary part, that is (z) = y. Given a square matrix F , we define with σ(F ) the spectrum of F and with p F (λ) its characteristic polynomial, namely p F (λ) = det(λI -F ). We denote with I i , or simply I, an identity matrix of dimension i ∈ Z >0 . We denote with 0 i,j a zero matrix of dimension i × j, with i, j ∈ Z >0 . We denote a triplet in prime form or simply (A,B,C) in the case of no dimension ambiguity), matrices of the form

(A i , B i , C i ) of dimension i ∈ Z >0 , (
A i = 0 i-1,1 I i-1 0 0 1,i-1 , B i = 0 i-1,1 1 , C T i = 1 0 i-1,1 . (1)

II. PRELIMINARIES

In this work we consider nonlinear autonomous systems of the form ẋ(t) = Ax(t) + Bϕ(x(t)) , y(t) = Cx(t) + ν(t) , [START_REF] Besanc ¸on | Nonlinear observers and applications[END_REF] where x ∈ R n is the state, n ∈ Z >0 , y ∈ R is the measured output, ν : R ≥0 → R is some (bounded) measurement noise, (A, B, C) is a triplet in prime form, see (1), of dimension n and the function ϕ(•) is at least locally Lipschitz. Note that the class (2) represents a fairly general class of systems (e.g. all linear observable and controllable dynamics) due to the general function ϕ(•). Moreover, we recall that any observable time-invariant autonomous nonlinear system can be written in the form (2) after a suitable change of coordinates, see [START_REF] Gauthier | Deterministic observation theory and applications[END_REF]- [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF] and references therein. We suppose that the state x of (2) evolves in a given compact set X ⊂ R n for all t ≥ 0. In the rest of the paper we will drop the time-dependency (t) on the variable when not useful. High-gain observers for system (2) can be designed as

ẋ = Ax + Bϕ s (x) + D k L(y -C x) (3)
where x ∈ R n is the state of the observer,

D k = diag(k, . . . , k n ),
where k ≥ 1 is a real number usually denoted as high-gain parameter, L = col( 1 , . . . , n ) is the Luenberger gain to be chosen such that (A-LC) is Hurwitz, and ϕ s is a globally bounded and locally Lipschitz function that agrees with ϕ on X, namely ϕ s (x) = ϕ(x) for all x ∈ X and there exists a number φ > 0 such that |ϕ s (x)| ≤ φ for all x ∈ R n . For instance, ϕ can be designed by saturating the function ϕ outside the compact set X. By using the fact that the matrix (A -LC) is Hurwitz, it can be shown that, for a k sufficiently large, the estimation error x -x satisfies the following bound

|x(t) -x(t)| ≤ k n-1 exp(-a 1 kt)a 2 + a 3 k n-1 sup t≥0 |ν(t)| (4)
where a i > 0, i = 1, 2, 3 are numbers independent of k. Bound (4) highlights that the estimation error converges asymptotically to zero when the measurement noise is not present. Moreover, observer (3) has good asymptotic properties with respect to the measurement noise (they can also be denoted as ISS-gain). On the other hand, the gain between the measurement noise and the estimation error is in general proportional to k n-1 showing that the faster is the observer, the worst will be in general the performances in the presence of measurement noise. As shown in [START_REF] Astolfi | Sensitivity to High-Frequency Measurement Noise of Nonlinear High-Gain Observers[END_REF], the low-pass filter behaviour of the high-gain observer (3) can be captured in case the measurement noise ν(t) is modelled by a finite Fourier series, namely

ν(t) := N i=1 ν i sin ω i t + φ i , (5)
for some N > 0, where the parameter > 0 parametrizes the basic frequencies ω i while ν i > 0 and φ i characterize the amplitude and the phase of the single components, respectively. The measurement noise (5) has a high-frequency behaviour for small enough. In this case, the asymptotic estimation error in (4) can be computed as

lim t→∞ |x(t) -x(t)| ≤ a 4 k n sup t≥0 |ν(t)| ≤ a 4 k n sup t≥0 i ν i (6)
where a 4 > 0 is a number independent of k. It is readily seen that the bound ( 6) catches the low-pass filter behaviour of the high-gain observer (3) since

lim →0 lim t→∞ |x(t) -x(t)| = 0 .
On the other hand, bound [START_REF] Astolfi | Sensitivity to High-Frequency Measurement Noise of Nonlinear High-Gain Observers[END_REF] highlights that the asymptotic gain at high-frequencies is proportional to k n and not to k n-1 . Nevertheless, as showed in [START_REF] Astolfi | Peaking-free low-power high-gain observers[END_REF], it can be shown that augmenting the relative degree between the measured output and the state estimations, may improve the asymptotic estimation [START_REF] Astolfi | Sensitivity to High-Frequency Measurement Noise of Nonlinear High-Gain Observers[END_REF]. For this reason, in this work, we propose to redesign the observer (3) by augmenting the relative degree between the measured output and the estimates. This task is accomplished by filtering the output injection term through a dynamical linear filter.

III. MAIN RESULTS

A. Observer structure

The high-gain observer (3) can be represented by a blockdiagram scheme as in Figure 1. The main objective of this work is to modify observer (3) by adding a suitable filter between the output injection term and the observer, as in Figure 2. In particular, in this section, we study a high-gain observer with filtering redesign of the form

ẋi = xi+1 + i η i , i = 1, . . . , n -1 ẋn = ϕ s (x) + n η n , η1 = -kαη 1 + k 2 β(y -x1 ), ηj = -kαη j + k 2 βη j-1 , j = 2, . . . , n, (7)
where (x, η) ∈ R 2n is the extended state, ϕ s is chosen as for the standard high-gain observer (3), and α, β, 1 , . . . , n are positive coefficients to be chosen. We denote with X × E ⊂ R n × R n the compact set of initial conditions of [START_REF] Ahrens | High-gain observer in the presence of measurement noise: a switched-gain approach[END_REF]. By inspecting the η-dynamics in [START_REF] Ahrens | High-gain observer in the presence of measurement noise: a switched-gain approach[END_REF], it can be noticed that η i plays the role of being a filtered version of β α k i (y -x1 ), thus recovering asymptotically the structure of the standard highgain observer (3) for low frequencies (in particular when α = β). In the following we will also use the forthcoming compact notation , arising from the fact that A is in the prime form [START_REF] Gauthier | Deterministic observation theory and applications[END_REF],

ẋ = Ax + Bϕ s (x) + diag(L)η, η = -k(αI -kβA T )η + βk 2 C T (y -C x), (8)
where the Luenberger gain L is defined as in [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF]. While in the standard high-gain observer (3), the coefficients 1 , . . . , n need to satisfy the condition (A -LC) Hurwitz, a more stringent condition is required for [START_REF] Sanfelice | On the performance of high-gain observers with gain adaptation under measurement noise[END_REF], as specified in the forthcoming section.

B. A preliminary Lemma

Let S α,β ⊂ C <0 be the following set

S α,β := (x + iy) ∈ C : α 2 x + βy 2 < 0 , (9)
whose peculiar parabolic boundary is represented in Figures 3 and 5. We have then the following lemma instrumental to the main result of this section.

Lemma 1 Let two positive reals α, β be fixed and consider the following matrix

M =   A diag(L) -βC T C -αI + βA T   . ( 10
)
The following holds:

(i) The characteristic polynomial of M satisfies p M (λ) = β n p (A-LC) λ(λ+α) β . ( 11
) (ii) If L is chosen such that σ(A -LC) ⊂ S α,β
, with S α,β defined in (9), then both (A -LC) and M are Hurwitz.

The proof of Lemma 1 is not detailed here due to space limits. The sketch of the proof is as follows. Relation [START_REF] Astolfi | Uniting local and global observers for the state estimation of nonlinear continuous-time systems[END_REF] expresses the characteristic polynomial of M , of degree 2n, in terms of the characteristic polynomial of (A -LC), of degree n, and parameters α and β. This relation can be obtained by iterative and suitable row and column expansions of the determinant det(M -λI) and using the structure of M . The proof of item (ii) of Lemma 1 exploits equation [START_REF] Astolfi | Uniting local and global observers for the state estimation of nonlinear continuous-time systems[END_REF], that is the link between the roots of p M (•) and p (A-LC) (•). Roughly speaking, we have λ ∈ σ(M ) if and only if

λ(λ+α) β = µ ∈ σ(A -LC).
The set S α,β defined by ( 9) is built as the intersection of two sets: the set of µ ∈ σ(A-LC) such that (µ) < 0 and in addition the set of µ ∈ σ(A-LC) such that both roots λ of the equation λ(λ + α) = βµ verify (λ) < 0. We obtain the conclusion that if σ(A -LC) ⊂ S α,β , then both (A -LC) and M are Hurwitz.

Lemma 1 requires further comments. The design of L such that both matrices (A -LC) and M be Hurwitz will start by selecting the spectrum σ(A -LC) ⊂ S α,β and in a second step by computing L related to this desired spectrum by exploiting the companion structure of matrix (A -LC). This procedure is detailed in Remark 1 at the end of this section.

We provide now two numerical examples illustrating the main features of Lemma 1. Numerical example 1. Consider the case of a system of dimension n = 8 and fix α = 0.1 and β = 0.1. We select in this first example L such that the eigenvalues of (A -LC) lay inside the set S α,β defined in [START_REF] Boizot | An adaptive high-gain observer for nonlinear systems[END_REF]. In particular we choose σ(A-LC) = {-2.5±0.3i, -2, -1.3, -1±0.2i, -0.1, -0.4}.

Figure 3 shows the placement of the eigenvalues (A -LC) and the parabola

α 2 (µ) + β 2 (µ) < 0 , (12)
corresponding to the boundary of set S α,β . The eigenvalues of the closed-loop matrix M in [START_REF] Boker | Nonlinear observers comprising highgain observers and extended Kalman filters[END_REF] are shown in Figure 4.

It can be noticed that all the 16 eigenvalues have strictly negative real part and that they are symmetric with respect to the vertical axis (λ) = -α/2 = -0.05, highlighted in red in Figure 4. This symmetric placement of the eigenvalues λ of M is a direct consequence of property (i) of Lemma 1.

Numerical example 2. We select, as in the previous Example 1, n = 8, α = 0.1 and β = 0.1. We select now L such that two eigenvalues of (A-LC) lay outside the set S α,β defined in [START_REF] Boizot | An adaptive high-gain observer for nonlinear systems[END_REF]. In particular we choose

σ(A-LC) = {-2.5±0.3i, -2, -1.3, -1±0.6i, -0.1, -0.4}.
Figure 5 shows the placement of the eigenvalues (A -LC) and the parabola [START_REF] Astolfi | A High-Gain Nonlinear Observer with Limited Gain Power[END_REF]. It can be noticed that the eigenvalues in -1 ± 0.6 are not inside the parabola defined by S α,β . These two defective eigenvalues are highlighted in magenta squares. The eigenvalues of the closed-loop matrix M in (10) are shown in Figure 6. It can be noticed that now 2 eigenvalues have positive real part, namely they are unstable. These eigenvalues are highlighted in magenta in Figure 6. Nevertheless, it can be remarked that the symmetry with respect to the vertical axis (λ) = -α/2 = -0.05, highlighted in red in Figure 6, is still respected.

C. Asymptotic properties of the estimation error

By direct application of Lemma 1 of the previous section, we obtain the following result. Theorem 1 Consider system (2) and observer [START_REF] Ahrens | High-gain observer in the presence of measurement noise: a switched-gain approach[END_REF]. Let α, β > 0 be fixed and let L be chosen such that σ(A-LC) ⊂ S α,β . Then, there exists a real number k * ≥ 1, such that, for any k > k * , the following bound holds

|x(t) -x(t)| ≤ k n-1 exp(-b 1 kt)b 2 + b 3 k n-1 sup t≥0 |ν(t)| (13
) for all t ≥ 0, for all initial conditions (x(0), x(0), η(0)) ∈ X × X × E and for some constants b i > 0, i = 1, 2, 3 independent of k.

Proof. By considering the change of coordinates e := kD -1 k (x -x), η := D -1 k η, and using

D -1 k A = kAD -1 k and kD -1 k A T = A T D -1 k , system (7) reads as ė = kAe + B∆(e, x) + kdiag(L)η η = -k(αI -βA T )η -kβC T Ce + βkC T ν (14)
where ∆(e, x) is computed as

∆(e, x) := k -(n-1) [ϕ s (D k k -1 e + x) -ϕ s (x)]. (15)
By setting ε := col(e, η), system ( 14) can be compactly written ε = kM ε + B∆(e, x) + kβ Cν,

where M is given in [START_REF] Boker | Nonlinear observers comprising highgain observers and extended Kalman filters[END_REF], B := col(B, 0), C = col(0, C T ). Since ϕ s is globally bounded and ϕ, ϕ s are locally Lipschitz, we have that |∆(e, x)| ≤ δ|e| ≤ δ|ε| for some δ > 0 for all x ∈ X and all e ∈ R n . As a consequence, since the matrix M is Hurwitz by construction (see in particular Lemma 1), by using the Lyapunvo function V = ε T P ε, with P = P T > 0 solution of P M + M T P = -I, and by applying standard Lyapunov arguments (see for instance [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF] or [START_REF] Astolfi | A High-Gain Nonlinear Observer with Limited Gain Power[END_REF]), which are here omitted for the sake of compactness, we can conclude that the following inequality

|ε(t)| ≤ exp(-b 1 kt)b 2 + b 3 sup t≥0 |ν(t)|,
holds for all t ≥ 0, for all k > k * , where k * ≥ 1 is large enough, and for some b i > 0, i = 1, 2, 3, independent of the parameter k. Finally, by using the previous bound and by recalling the inequality |x -x| ≤ k n-1 |e| ≤ k n-1 |ε| we obtain the bound [START_REF] Teel | Further variants of the Astolfi/Marconi high-gain observer[END_REF], which concludes the proof.

Remark 1 A nice feature to be highlighted is the design of observer [START_REF] Ahrens | High-gain observer in the presence of measurement noise: a switched-gain approach[END_REF] and in particular how to choose the coefficients α, β, 1 , . . . , n characterizing the gains of the observer. In particular, in view of Lemma 1, one can simply follow these steps:

• choose any α, β > 0;

• choose n stable poles λ i (that can be real or complex conjugate) lying inside S αβ of (9); • compute the resulting characteristic polynomial n i=1 (λ -λ i ) = λ n + a 1 λ n-1 + . . . + a n-1 λ + a n ; note that, in light of the Routh-Hurwitz necessary condition for the stability of a polynomial, we have a i > 0 for all i = 1, . . . , n;

• assign i = a i for all i = 1, . . . , n. As it can be noticed, the procedure is substantially the one used to select the gain of a standard high-gain observer, but with the additional constraint that the poles of (A -LC) lay inside S α,β ⊂ C <0 .

The proposed observer [START_REF] Ahrens | High-gain observer in the presence of measurement noise: a switched-gain approach[END_REF] ensures asymptotic convergence to zero of the estimation error |x -x|. Even if the bound (13) has the same form as (4), the presence of the additional dynamics η may improve the performances of the overall observer in the presence of measurement noise with respect to the standard high-gain observer [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF]. A detailed characterization of these performances is out of the scope of this preliminary work. However, we may note that the relative degree between the measured output y and the estimation x is in general at least 2, so better performances in presence of high-frequency measurement noise are in general expected. In particular, by following the computation in [START_REF] Astolfi | Sensitivity to High-Frequency Measurement Noise of Nonlinear High-Gain Observers[END_REF], it is expected that when the measurement noise ν(t) is modelled as in [START_REF] Ball | High-gain observer tracking performance in the presence of measurement noise[END_REF], the asymptotic behaviour of the estimation error for [START_REF] Ahrens | High-gain observer in the presence of measurement noise: a switched-gain approach[END_REF] can be characterized by

lim t→∞ |x(t) -x(t)| ≤ 2 b 4 k n sup t≥0 |ν(t)|, ( 17
)
for some b 4 > 0 independent of k. We remark that in view of the special structure of the dynamical filter in [START_REF] Ahrens | High-gain observer in the presence of measurement noise: a switched-gain approach[END_REF], the bound ( 17) could be refined by inspecting all its components xi -x i , i = 1, . . . , n, as done for instance in [START_REF] Astolfi | Sensitivity to High-Frequency Measurement Noise of Nonlinear High-Gain Observers[END_REF], [START_REF] Astolfi | Peaking-free low-power high-gain observers[END_REF]. However, because of space reasons, in this work we limit ourselves to show the effectiveness of the scheme in the numerical example in Section V. Finally, we remark that the proposed structure (7) has the desirable feature of having the high-gain parameter k which is powered up to 2 regardless of the dimension of the system state n. On the contrary, standard high-gain observers embed coefficients which are proportional to k n . This feature may be of particular interest in case of a large state-dimension n for numerical implementation purposes, as already highlighted in [START_REF] Astolfi | A High-Gain Nonlinear Observer with Limited Gain Power[END_REF]- [START_REF] Khalil | Cascade high-gain observers in output feedback control[END_REF].

IV. A GENERAL FORMULATION

By following the general scheme of Figure 2, it can be noticed that the observer structure (8) presented in Section III can be further generalized as follows

ẋ = Ax + Bϕ s (x) + D k N Hη, η = kF η + kG(y -C x), (18)
where x ∈ R n is the estimate of the state x of (2), η ∈ R m is the state of the filter, with m ∈ Z >0 . We denote with X × E ⊂ R n × R m the compact set of initial conditions of [START_REF] Khalil | Analysis of the Use of Low-Pass Filters with High-Gain Observers[END_REF]. Matrices N, F, G, H are now degrees of freedom to be properly chosen. It is readily seen that the structure k , H = diag(L), F = (αI -kβA T ) and G = βkC T . By analysing the dynamics of system (3) and the observer (18), we can establish the following result. Theorem 2 Consider system (2) and observer [START_REF] Khalil | Analysis of the Use of Low-Pass Filters with High-Gain Observers[END_REF]. Let N, F, G, H be chosen such that the following matrix

Q := A N H -GC F (19)
is Hurwitz. Then, there exists a k * ≥ 1, such that, for any k > k * , the following bound holds

|x(t) -x(t)| ≤ k n-1 exp(-b 1 kt)b 2 + b 3 k n-1 sup t≥0 |ν(t)|, (20)
for all t ≥ 0, for all initial conditions (x(0), x(0), η(0)) ∈ X × X × E and for some constants b i > 0, i = 1, 2, 3 independent of k. where B := col(B, 0), Ḡ = col(0, G) and ∆ is computed as in [START_REF] Astolfi | Peaking-free low-power high-gain observers[END_REF]. The end of the proof follows by following the same steps used in the proof of Theorem 1, and by recalling that the matrix Q is Hurwitz by assumption.

Theorem 2 states that observer [START_REF] Khalil | Analysis of the Use of Low-Pass Filters with High-Gain Observers[END_REF] guarantees asymptotic estimation of system (2) in nominal condition and moreover that the estimation error is stable with respect to the measurement noise. This result may be seen as a generalization of Theorem 1. Furthermore, in view of the mild condition Q Hurwitz, we believe that all the degrees of freedom, namely the dimension m of the filter and the matrices F, G, N, H, could be selected in order to improve the behaviour of the observer in presence of measurement noise with some optimality criteria (deterministic or stochastic).

Remark 2 In this work we have not investigated the use of the new observer structure [START_REF] Khalil | Analysis of the Use of Low-Pass Filters with High-Gain Observers[END_REF] in output feedback stabilization. However, in light of the properties highlighted in (20), it is expected that observers of the form (18) can be used in place of standard high-gain observers (3) without loss of generality since the same main features are retained (arbitrarily fast exponential convergence of the estimation error and input-to-state stability properties with respect to the measurement noise). Finally, we remark that also some robustness properties with respect to model uncertainties of the function ϕ of the plant (2) are also expected, as in the case of standard high-gain observers.

V. SIMULATION

We consider the following nonlinear system

ẋ = A 4 x + B 4 ϕ(x), y = C 4 x + ν (22)
where x ∈ R 4 is the state, (A 4 , B 4 , C 4 ) is a triplet in prime form of dimension 4, and the function ϕ(x) is chosen as ϕ(x) = -x 4 -5.2x 3 -2.7x 2 + 4.5(x 2 1 -1) . This system, for initial conditions x = col(0.9, 0, 0, 0), has a chaotic attractor and bounded trajectories for all t ≥ 0, see [START_REF] Chlouverakis | Chaotic hyperjerk systems[END_REF]. Furthermore, it can be numerically verified that solutions converge to a set where |x i (t)| ≤ 6 and |ϕ(x(t))| < 25 for all t ≥ 0. The measurement noise is modelled as ν(t) = ν sin(ωt) with ν = 0.1 and ω ≥ 50.

We implement the filtered high-gain observer (7) by selecting ϕ s (x) = sat 25 (ϕ(x)) where sat r (s) = max(-r, min(r, s)), α = 3, β = 1 and the poles of (A-LC) are placed in (-1±i, -2±i), which corresponds to selecting L = col [START_REF] Astolfi | Sensitivity to High-Frequency Measurement Noise of Nonlinear High-Gain Observers[END_REF][START_REF] Astolfi | Peaking-free low-power high-gain observers[END_REF][START_REF] Khalil | Analysis of the Use of Low-Pass Filters with High-Gain Observers[END_REF][START_REF] Boker | Nonlinear observers comprising highgain observers and extended Kalman filters[END_REF]. It can be verified numerically that the poles of M are complex conjugates with real part placed between -2.7 and -0.3. Asymptotic estimation is achieved in nominal conditions, namely when there is no measurement noise, for k ≥ 5.

As a comparison, we run the simulation in the same conditions while considering a standard high-gain observer of the form (3). In this case, L is taken as before, but we choose k = 4 which is large enough to guarantee convergence of the observer in nominal conditions, namely when ν = 0. Figure 7 shows convergence of the estimation errors |x(t) -x(t)| of the two observers. For both observers the estimation error convergences asymptotically to zero with a convergence rate which is of the same order. It can be noticed that the presence of the filters in observer [START_REF] Ahrens | High-gain observer in the presence of measurement noise: a switched-gain approach[END_REF] attenuates the peaking phenomena during the transient.

Table I contains the data concerning the normalised asymptotic estimation errors

xi a := lim sup t→∞ |x i (t) -x i (t)| ν (23)
for various values of ω when k = 10. Improvement of the performances is achieved when ω is taken higher. On the other hand it can be seen that the effect of the measurement on the last components of the estimation error is not amplified as in the case of standard high-gain observer. Finally, Table II shows the effect of the measurement noise on the asymptotic estimation errors (values are normalised with respect to ν as in Table I). It can be noticed that the measurement noise is amplified on the lower components and in general the estimation error provided by the filtered high-gain observer [START_REF] Ahrens | High-gain observer in the presence of measurement noise: a switched-gain approach[END_REF] has lower bounds (see Table I) even though the high-gain parameter is larger. An exact comparison between the two observers is however very hard to address since the dynamics are different because of the different dimension and eigenvalues of the closed-loop rescaled matrices (A -LC) and M .

VI. CONCLUSION

We proposed to modify the standard high-gain observer structure by adding a dynamical filter processing the output injection term. We proved that the obtained observer inherits the asymptotic behaviour of the high-gain observer, namely asymptotic estimation can be achieved in nominal conditions, and input-to-state properties with respect to measurement noise are retained. We expect that the effect of augmenting the relative degree between estimates and measurements, and the choice of the dynamical filter, may improve the sensitivity with respect to measurement noise. This fact is confirmed by simulations although a rigorous proof is not provided. A particular choice of the filter allows to implement coefficients that are proportional to powers of the high-gain parameter equal to one or two, thus resulting in numerical benefits when the system's dimension is large or fast convergence is sought for (see [START_REF] Astolfi | A High-Gain Nonlinear Observer with Limited Gain Power[END_REF], [START_REF] Khalil | Cascade high-gain observers in output feedback control[END_REF]).

This work proposes a different route for the problem of state estimation of nonlinear systems in the presence of measurement noise, which mainly combines a Luenberger observer with a dynamical filter in a feedback form (and not in cascade) and sufficient conditions for the stability of the composed observer are given. A full characterization of the behaviour of the proposed technique in presence of measurement noise and optimality-based tuning for both the gain and the parameters of the dynamical filter remains an open question for future research.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Standard observer scheme.

Fig. 3 :Fig. 4 :

 34 Fig.3: Numerical example 1. Choice of the stable eigenvalues of (A -LC).

Fig. 5 :Fig. 6 :

 56 Fig.5: Numerical example 2. Choice of the stable eigenvalues of (A -LC).

Proof.

  By applying the change of coordinates e := kD -1 k (x -x) and by denoting ε := col(e, η), system (18) reads ε = kQε + B∆(e, x) + k Ḡν (21)

Fig. 7 :

 7 Fig. 7: Convergence of the estimation error |x(t) -x(t)|. Dotted blue line: standard high-gain observer (3). Red line: high-gain observer with filtering redesign (7).

TABLE I :

 I Normalized asymptotic estimation error (23) of the filtered high-gain observer[START_REF] Ahrens | High-gain observer in the presence of measurement noise: a switched-gain approach[END_REF].

		ω = 50	ω = 100 ω = 300
	x1 a	2	0.6	0.07
	x2 a	10	1.4	0.07
	x3 a	22	1.6	0.02
	x4 a	18	0.8	0.01

TABLE II :

 II Normalized asymptotic estimation error (23) of the standard high-gain observer (3).

		ω = 50	ω = 100 ω = 300
	x1 a	5	2.5	0.9
	x2 a	50		8
	a	230	120	40
	x4 a	500	260	90


Anne Fournier
email: anne.fournier@ird.fr

Divination "with plants" in West Africa

Keywords: Burkina Faso, Volta River Basin cultural area, Sèmè society, bush spirits, symbolism of plants, ritual practices

The Sèmè make particularly abundant use of plants in the field of divination. Diviners practice their art with the assistance of a bush spirit that comes to live with them at home. Plants serve as material support for settling this bush spirit in the diviner's home and as powerful objects on the divination scene. Each species of plant plays one or several specific roles. This article is based on extensive field research, and the analysis of the meaning of the plants used in divination (names in the Sèmè language, ritual use). It describes the role of plant species and their symbolic roles in settling the bush spirit in the diviner's home and during the divination process. Although similar to those of other societies located in the Volta River Basin, the Sèmè divination practices show some original aspects.

Introduction

Divination is a well-established institution that informs and stimulates ritual acts such as births, marriages, deaths, and initiations in unindustrialized societies (de Surgy 2013). In Africa, specifically, divination is widespread, with practices that are both modified or inherited directly from the past [START_REF] Peek | The study of divination, present and past[END_REF]. It is practiced by people of all ages and socio-professional categories and is based on a complex set of representations relating to social and biological reproduction, the village territory, and invisible beings such as the creator, ancestors, bush spirits and other deities or supernatural forces [START_REF] Surgy (de) | La divination par les huit cordelettes chez les Mwaba-Guma. 1 Esquisse de leurs croyances religieuses[END_REF][START_REF] Capron | Le Grand jeu : le mythe de creátion chez les Bwa-Pwesya, Burkina Faso, 1986-1987[END_REF][START_REF] Fournier | Setting up the first components of the person and its anchoring to the territory among the Seme of Burkina Faso: "services rendered by ecosystems[END_REF].

As noted by [START_REF] Peek | Reality reviewed: dynamics of African divination[END_REF], "The technical forms of divination are legion, and usually involve a randomizing agent or act (throwing or rubbing objects, animals or parts of animals), or the inducement of trance and intuition, which then leads to complex interpretations or calculations." In the Volta River Basin of West Africa, divination using a wand is widespread, probably an ancient divination process and it is also common to throw cowries (the marine species Cypraea moneta, formerly the currency in West Africa) (Liberski-Bagnoud 2012).

In the Volta River Basin cultural area of West Africa, diviners are sought out when people have disturbing dreams or repeated failures and misfortunes. In their eyes, this may indicate that one of the many invisible entities that make up their pantheons is sending them a sign that something must be done (prayer, sacrifice, other ritual acts). In addition, they also seek the diviner as a preventive measure before undertaking an important act. The most common way to become a diviner is to enter into a union with a supernatural entity. This union is sealed by the construction of an altar for the supernatural entity, often a bush spirit, at the person's home [START_REF] Rouville (de) | Les cérémonies d'initiation du bur[END_REF]de Surgy 1986: 61;Fortes 1987:13;[START_REF] Blier | Diviners as alienists and annunciators among the Batammaliba of Togo[END_REF][START_REF] Dugast | Apparitions et figurations de l'invisible chez les Bwaba du Burkina Faso : 2. Les métamorphoses de l'autel du devin[END_REF]. Bush spirits (or "génies" in the French translation from local languages) are invisible beings endowed with the power of transformation. They live their lives almost like humans, but their bodies and behavior show distortions and inversions compared with those of humans [START_REF] Hamberger | Traces des génies[END_REF]. This article demonstrates that plants are essential material supports among the Sèmè of Burkina Faso (West Africa) for building the relationship between a diviner and a bush spirit and in conducting the divination sessions. It explains the symbolism of the plants that are used and presents the beliefs on which this symbolism is based. As shown in the final discussion section, these beliefs constitute the Sèmè version of a core of representations that are shared by the societies of the Volta River Basin.

Study Area and Methods

The Sèmè people of West Africa are concentrated today around the town of Orodara (Figure 1). According to historians, linguists, and anthropologists, their society was constituted in the 17th century from a small group of people originating from Côte d'Ivoire [START_REF] Person | Des kru en Haute-Volta[END_REF][START_REF] Prost | Contribution à l'étude des langues voltaiques[END_REF][START_REF] Schwartz | Tradition et changements dans la société guéré (Côte d'Ivoire). Orstom (IRD)[END_REF]Schwartz : 20, 1993:116-119):116-119). There is no centralized power above that of the village; the family is patrilineal and residence is virilocal. Their social structure is divided into four social status groups: the farmers, the gravediggers, the blacksmiths, and the griots. Even though Christianity and Islam are today very widespread in Burkina Faso [START_REF] Langewiesche | Mobilité religieuse : changements religieux au Burkina Faso[END_REF], the "traditional" religion organized around the dwo cult remains very much alive around Orodara [START_REF] Trost | Tradition und Veränderung der Gesellschaftsstruktur bei den Tussian, Burkina Faso[END_REF]. Studies have been conducted on the natural environments and their management, vernacular names of plants and their main uses [START_REF] Bene | Végétation naturelle et occupation des terres au Burkina Faso (Afrique de l'ouest). Cinq décennies de changement dans un terroir du pays sèmè[END_REF]Fournier et al. 2014;[START_REF] Boyd | Une base de données informatisée transdisciplinaire de la flore : un outil pour l'étude du lien nature-société[END_REF]. However, there is little documentation about the Sèmè culture and ethnography and even less about their divination practices.

The study is based on a field survey of local specialists known for their knowledge of divination and bush spirits and/or plants and their medicinal and magical uses. The author collected data on divination in Orodara and nearby Sèmè villages between 2010 and 2017. Over a hundred semi-structured interviews were conducted on divination, bush spirits, plants, and their magical and other uses with people over the age of 50 (diviners, healers, heads of families and dwo cult officials). Followers of bush spirits and other people reported their experiences as novices, initiators, or observers. Translation into French, was provided by a native speaker knowledgeable of Sèmè traditions. Rich comments were provided by the Sèmè about the vocabulary used in divination and the plant names (Supplementary Table I andII). They also explained the symbolic meanings of the ritual uses of these plants. Recordings made by the author during these interviews also contributed to a study of the Sèmè language conducted by linguists (RADICEL-K program, http://radicel-k.huma-num.fr/).

Divination method

Divination takes place in the house of the bush spirit or in the courtyard of the diviner or of the client, who does not explain the reason for his coming. In the divination process, the diviner asks the bush spirit questions, translates the answers and then interprets them for the client. The bush spirit's "words" are materialized by 6 cowries on a flat black stone and by the movements of a forked wand that is rubbed on it. After taking out his materials from his bag (Supplementary Figure 1), the diviner performs gestures of salutation to God and cleans the stone ritually (Supplementary Video 6 s.). Then he checks, from a first series of throws, that each of the cowries is ready to speak and that the bush spirit is present. He greets the cowries lined up on the stone with small plant "fetishes" (Table 3), which he then deposits on the stone or near it (Supplementary Figures 1,2, Video 1 ). The diviner and the client each hold a branch of the forked wand (Supplementary Video, 1 min 33 s) and while rubbing the wand back and forth on the stone, the diviner asks questions. When his words are true, the wand slides, but when he is mistaken, it stops. Every answer is checked by throwing the cowries (Supplementary Video: 2 min 27 s).

Rites for settling the Bush Spirit

The union of a man with a bush spirit requires that he be "initiated to this bush spirit" and it is the latter, not the man, that makes the decision. A Sèmè person who experiences problems or misfortunes will react by consulting a diviner, who may explain that a bush spirit has decided to settle in the person's home. This person must then become part of the group of followers of this "bush spirit." The details of the welcome ceremony for the bush spirit are identified by divination: the tyéén no sacred site ("mother of the bush spirit") where it lives, the day that would be favorable and the plant species that are suitable for making its altars. A small house is then built in the family courtyard and a branch of the tyéén-kaar tree (Tables 1,2) is plunged into the top of its straw roof (Supplementary Figure 3). When the day comes, a group of male followers of this spirit go to the tyéén no site to perform a blood sacrifice. Plant material that will be used to settle the bush spirit in the novice's home is collected along the way, especially two bunches of roots that must be presented at the tyéén no site, and one of them is partially burned. As any male novice is potentially a diviner, the items necessary for divination are also presented to the site and activated. Each follower places a crown of the baarn-byel liana on his head and the group goes back to the village.

Back at the house of the novice, the initiators will "piquer le génie" [stake down the bush spirit], a French translation by the Sèmè of the special expression kool kpé. As there is actually a pair of bush spirits, initiators build a first portico about sixty centimeters high outside the family courtyard, which represents the male spirit. They plant two forked pickets in the ground and place some crosspieces on them. Some of the lianas that were used as crowns are wrapped around the crosspieces and then a branch of tyéén-kaar is placed on the lianas. After performing various sacrifices, they build a second portico inside the house representing the female spirit (Supplementary Figure 4). Under this altar, up to three small pots are placed in hollows in the ground and the other bunch of roots (not burned) is deposited in them. A small forked root, which was gathered separately and will form part of the materials used in divination, is placed across one of these pots. The remaining crowns are wrapped around the crosspieces and a branch of tyéén-kaar is deposited on them. Other sacrifices are made on the altar and the pots are filled with drinking water for the bush spirit. From now on, they must never remain empty, because the bush spirit is supposed to be living in the house.

During the celebration for the arrival of the bush spirit, guests make small gifts of money or cowries. A woman chosen by divination receives them in a small palm leaf winnowing basket, which she was required to provide. She will eventually keep these gifts for herself, but must first place them on the altar in the house. The initiators hang the winnowing basket under the altar and sometimes add a ritual marriage broom made of grass. They finely grind the bunch of roots already partially charred, swallow a small amount of the powder, and divide up the rest to take home with them. The ceremony ends with a festive meal, prepared and served separately for the followers and the novice on one hand and the family, neighbors and friends on the other. A meal is also served to the bush spirit in its house, either on the baarn-byel lianas on the altar, or in small pieces of broken calabashes arranged on the ground in front of the pots. Generally, the bush spirit requires special food to be served to it, prepared by mixing with water some powdered seeds (white or more rarely red) of the fel plant. Unlike the kind of porridge made of Bambara nuts commonly eaten by the Sèmè that it seems to imitate, this dish is left raw. If they were unable to find the seeds of the fel plant, they give the bush spirit the same meal as the followers but they ask to be forgiven when presenting it with the shel "fetish" (an object endowed with power and used for magico-religious practices). For 5 to 10 days, the bush spirit's house will then remain ritually closed. During this period, the novice procures, with the assistance of his initiators, objects that may be requested by the bush spirit and plants that must be added to the roots and he begins to learn divination.

After a probationary period of three years, rites identical to those for the reception of the bush spirit are held, except that no more plant material is collected and no one wears a crown.

The meal that is served again must include exactly the same dishes for the bush spirit. This stage, called tyéén no lo, involves visiting the tyéén no sacred site again and making sacrifices to it. If the novice had been involved in witchcraft, the bush spirit would have killed him: to pass this milestone means that the spirit has approved their alliance. The novice can then make sacrifices to the tyéén no site without going through the older followers.

Catching and "building" the Bush Spirit

After the construction of a small house, the bush spirit is ritually displaced from the bush into this new dwelling and wooden effigies are built. Why do the Sèmè select specific plant species to perform this? They say with a touch of humor that, like Muslims who return from Mecca, we must wear a special costume to show what we are doing. The baarn byel crown, which suggests a cushion to carry the bush spirit, is worn when it is brought to the novice's home, but not at the confirmation ceremony, because the bush spirit has already moved in. To understand the winding of the liana around the pickets, it is necessary to examine the meaning of the name of this plant. The first etymological comments led to translating baarn-byel as "hare's porridge" [START_REF] Boyd | Une base de données informatisée transdisciplinaire de la flore : un outil pour l'étude du lien nature-société[END_REF], which seems absurd since the hare does not consume this toxic plant. Some elders explained that byèl indeed refers to the porridge that is still liquid before being boiled down and the residue that sticks to the pot. However, as a verb, byel describes the act of stirring a mixture of water and flour when thickening a millet porridge. By extension (and as a noun) it describes items consisting of ropes or threads that are inextricably entangled and disorderly. Hence, in this name, byel does not suggest a food use, but refers to the tangled appearance of the plant. This is a semiparasitic liana of greenish or orange color devoid of leaves (Table 1). When it begins to grow, this plant rises and wraps itself around a bush or a shrub, then it falls to the ground and there it spreads out in entangled clusters. Bush spirits use this plant as a rope, as humans do when they use it to tie the pickets for the bush spirit's altar. The hare is associated with this plant because when hunters are seeking game, small animals, especially hares, are chased into the clusters it forms. In addition, hunters fumigate their guns with baarn-byel to prevent game from escaping when it sees them. The Sèmè see the hare as the wild animal closest to humans because it lives in the bush, but feeds especially in the fields near their houses. For the Bobo (neighbors of the Sèmè), this behavior makes the hare an agent of liaison and transmission between humans and nature (Sanou 1993: 231). In short, baarn-byel is a magic rope that facilitates the capture of wild animals, especially the hare, an animal mediator between the bush and the village, hence its symbolic importance for bringing a bush spirit from the bush to the homes of humans.

The full name of the bush spirit's altar is tyéén kan, but it is often called tyéén for short.

Kan, "leg", "step" or "stride", also refers to the distance between a person's feet when standing up, the lower part of the human body from the waist and the forked handle of a slingshot and, by extension, to any bifurcated object. The Sèmè say that the wooden portico "is the bush spirit" or is "its legs." On the other hand, they specify that the bush spirit is in no way to be confused with the construction: at most it is seated upon it. It certainly is not shut up inside it and it circulates; this is why the diviner begins consultations by calling his bush spirit to ensure that it is present.

The species of tree used for building altars is chosen by divination from a closed list (komo, kukwal, wo and kpaal). In principle, one or two trees species are assigned to each tyéén no site, however, the bush spirit is free to demand a different species from among these four. These trees were once common and relatively abundant in the savannah. Komo, the most frequently used species today, is said to have been the first species used. The stories explaining the preference for particular species have almost all been forgotten. The few short and imprecise stories that could be collected resemble those explaining the clan "totems:" when a tree or animal has given protection against a dangerous animal or an enemy or has provided a supply of edible fruits during a famine, etc., an ancestor of the group decides that his descendants will "respect" the species. Totemic relations imply a food and use prohibition, while in the case of tyéén no sites, the use of the species is prescribed for the building of altars. One elder gave another explanation: "it may have simply been the tree under which the bush spirit rested when it was at home." This refers to a common mode of denomination among the Sèmè. The names of some families or subdistricts originate from that of the big tree under which (or near which) their founder settled.

Holding the Bush Spirit in Place

A group of plant species called tyéén-kaar is used recurrently in the rites of the Sèmè who possess their own botanical classification of these species. They commented on the words that make up their names and on the symbolism of the ritual use of these species. This led us to understand that tyéén-kaar serves to hold in place the bush spirit that assists the diviner.

With a combination of five terms, the Sèmè name three species of the Gardenia botanical genus (Table 2). They identify a female (G. erubescens) and a male form (G. ternifolia and G. aqualla). They describe the leaves of the first as wider and "well formed" and those of the second as being "cut off", like the ends of a leper's limbs, from their more defined contours. According to the linguistic database, kaar is the generic form (in the linguistic sense) that designates all three botanical species of Gardenia. However, the elders whom we consulted felt that the complete name is tyéén-kaar, to which could be added tiin (male) or gnéén (female). The rare few who distinguished between G. ternifolia and G. aqualla considered only the first to be male.

One of them proposed the name of tyéén-kaar bobo for G. aqualla, because this plant "serves no purpose" like the deaf-mute (bobo), who is said to be "useless". The male form would be used for ritual purposes only if the other could not be found.

The first term of tyéén-kaar is invariably identified as referring to a bush spirit, whereas various interpretations were given to kaar. The homophone kar (plural kaar) refers to various objects in the form of a ring or containing plants with protective properties. These include herbal amulets (bracelets, necklaces, belts) that the Sèmè once always wore. Kar also refers to metal bracelets and rings worn as amulets or deposited on altars dedicated to powers related to war, also known as kar. In a different context, kar is the name of the robust ring of climbing plants (kpéél or bwoo) that once held the thatched cone of the roof of houses (Supplementary Figure 3). The compound term bunkar (/in a group, kar/) refers to the thick ring placed inside and at the top of the conical frame formed by a bundle of Raphia palm stipes to maintain and consolidate it (Supplementary Figure 5). It was once always attached with the fibers of the bark of the tyaal tree. Tyaal fibers are also intertwined in the lower part of the frame where they form other rings (Supplementary Figures 6); their name (kaar-ja mon) contains the exact homophone of the term included in tyéén-kaar. According to the linguistic database of RADICEL-K program, kaar means "liana" and kar-bo refers to a clump of lianas, without indication whether kar is a generic name. According to our informants, a unique word appears in the vocabulary related to the roof and it means "liana," but they could not decide if it was kar or kaar. This could be the name of a particular species of liana (not identified) that grows near creeks. Finally, citing another imperfect homophone, someone went on to joke, "kaar, alone, is a scorpion, but it has nothing to do with the plant." However, to the extent that bush spirits punish those who break the rules by scorpion stings, this joke is worth mentioning.

In different Sèmè rites, the tyéén-kaar plant (Gardenia species see below) sometimes seems to indicate the meaning of separating things or beings from each other, and sometimes to bring them together. A branch of it stuck into the bush spirit's house signals a special place, but it also prevents the hens from sitting on the roof, just as the branch on the altar of the male bush spirit prevents goats from climbing on it or rubbing against it. A branch of tyéén-kaar is also usually seen on the ancestors' altar: it is said to prevent children from running around in there. At the ceremony that brings about the transformation of a dead person into an ancestor, this branch is always deposited on the altar after the stone that represents the new ancestor [START_REF] Fournier | Setting up the first components of the person and its anchoring to the territory among the Seme of Burkina Faso: "services rendered by ecosystems[END_REF].

When a hunter is threatened by game, he takes refuge under a tyéén-kaar tree, because the animal would never come to attack him there.

Sèmè plant connoisseurs explain that this plant likes to gather together, to unite. They point out that its branches catch the clothes of those who pass by it, as if to seize them. Thus, the main role of the tyéén-kaar branches deposited in the places mentioned above is to "assemble," "gather together," and "keep together" the humans and invisible beings concerned. Placing this branch means that the moral contract between the bush spirit and its host is sealed. We may suggest that this branch could also separate the domesticated spirit from its fellow bush spirits who remain in the bush.

To give high symbolic value to the Gardenia species seems characteristic of West African societies and especially of inhabitants of the Volta River Basin. They represent the dwo cult among the Bobo of Burkina Faso [START_REF] Sanou | La notion de parole chez les Bobo (étude ethnolinguistique)[END_REF] and are the seat of the vodou among the Ouatchi of Togo (Hamberger 2011: 436). Among the Winye of Burkina Faso, the Gardenia is the ancestor of the trees and it possesses, as among the Sèmè, powers of protection and assembly (Jacob, pers. comm. March 14, 2017). Among the Bwaba of Burkina Faso, they ward off thieves and drive the banished away from the village.

Creating Connections with the Bush

The Sèmè feel that roots are the most powerful parts of the plants; in the rite described, they use them to create or reinforce connections with the bush. The nature of the roots and the way they are gathered are some of the most secret aspects of the rites, but we were able to gather some information. The roots that are kept immersed in water in the pots (kool naan diir, in the plural) are from one or more of the four species dedicated to the tyéén no sacred sites. In some cases, they would be the same species as the pickets used for the altar, while in others, the choice would be made by divination. However, the mixture is apparently completed by other roots gathered near a termite mound or an anthill, places known to be haunted by the bush spirits. They may also be roots of plants that have grown at the market or across a road, places where everyone goes by. This enables them to "see and hear everything" which helps the diviner to detect the truth.

Followers gave concrete (but metaphorical) explanations for using these roots. Some said that trees only stand upright (ja) owing to their roots, hence some must be added to the pickets of the altar. For followers of certain sacred sites, pots and roots are always placed before building the altar because the roots necessarily come before the tree, whereas followers of other sacred sites place the pots after building the altars. The roots are placed in the pots while the altar is left visible, because in the bush the roots are hidden in the soil while the branches are "things on high." The Sèmè say that a man or a bush spirit, just like a tree, "only stands on its roots." It is a way of referring to the many "medications" (almost always root-based) with which they surround themselves for protection from attacks by witches. When a Sèmè talks about someone's roots, he is also referring to his network of relationships, his family, and all the financial and other support from which he may benefit. This is a representation shared by the societies of the Volta River Basin, such as for example, the Winye of Burkina Faso who also consider fetishes to be trees, with roots, like men and bush spirits (Jacob, pers. comm. March 14, 2017).

In addition, bush spirits, who do not mind drinking simple water when they turn into humans, refuse water in which there are no roots immersed when they assume their true form of beings from the bush. Roots transform this water, which becomes like the water of rivers (which always flow over roots), and make it suitable for consumption by bush spirits as they are. The mediating power of roots is confirmed in other uses of this water. The follower and his wife must from time to time rub it on their heads, faces and bodies as a greeting to the spirit and a request for its protection. If a follower is unable to do this for more than twenty days, he requests one of his brothers to ask for the forgiveness of the spirit by placing a shel fetish on the path he took when leaving. When babies cry or have a fever, it is understood to be the result of the fear caused by the bush spirits they see, an ability that older children and adults lose. They simply have to be rubbed with this water to stop the problem. These roots thus seem to have the power to introduce the bush into the village.

Kool naan dirn (in the singular) is a root, or sometimes a branch, of 5 to 10 centimeters long that may also be forked at one end. It is placed temporarily across one pot in the house and then forms part of the materials for divination (Supplementary Video: 1 min 10 s). It is always prominently displayed and is referred to during a divination. Diviners say that it "summarizes" (a tobra or yen) the roots placed in the pots and recalls the altar. Indeed, it is impossible to undertake divination without this root, which is the "door that provides access to divination." It "forces the truth to come out" or "directs" one towards the truth or "puts one on the right path" (a bré hlo gnaal taa). Only roots from among the four dedicated species of the tyéén no sites may be used, the most frequent being komo.

Each new follower receives a "divination tail" (sé-daar) (Supplementary Video: the small cloth bag at 1 min 33 s). Different groups of followers have their own way of making it, but a dough made of plant tubers seems to be the main component (and sometimes wild yams called kursana). The mixture may be sewn inside the skin of the testicle of the animal sacrificed to the tyéén no site, but there are other forms. The collective noun sé-daar bar mon describes various plants that form part of it, making it possible to refer to them without revealing their identity. On pain of being punished by the bush spirit, the follower must wear his séé-dar on him whenever he leaves home: the object signifies his bond with the bush spirit and protects him from threats originating from the invisible world.

The Plants of the Oracle

To set up the divination scene, the diviner manipulates plant parts and objects or substances of plant origin. The relation of these objects and plant parts to the bush spirits and the dwo cult must now be described.

During divination, a forked wand delivers the "words" of the bush spirit. This essential object is made from a tree taken from the bush (Table 3). Its wood must be very hard because if it breaks during divination, the diviner or the client will die. In the past, various species were used, including kpéél, but today only the more abundant thorny bush blibli is used. Bush spirits like it because they can hide in it and keep their belongings out of reach of humans.

When the diviner holds his divination in the house of the bush spirit, he lays the divination stone on a fixed stand made of adobe (Supplementary Video: 1 min 54). If he works elsewhere, he places it on a small portable support called kool gniné mon (/genie, to sit, thing/ literally, a thing for the genie to sit on), which must necessarily be made of the wood of the palmyra palm (Supplementary Figure 2, Video: 1 min 33 s).

When the followers ingest the powder made of the charred, ground roots before performing a divination, their words flow more smoothly and it helps them to "tell the truth," but the need for it decreases for seasoned specialists. The diviners of certain groups of followers use it to permeate the cloth cap of the horn in which the cowries are kept. By scrubbing the stone on which they throw the cowries with this cloth each morning or before each new divination, they cleanse it (a ta kukuur) ritually (Supplementary, Video: 8 to 20 s) and avoid clouding the work of the oracle. Another purifying option is to make the cowries "drink" this powder once and for all during the activation of the divination materials and then use a simple cloth. The diviners gave us to understand that wo (one of the dedicated species of the tyéén no sites) is often part of the mixture.

Setting up the divination scene requires calling the bush spirits with two metal bells and disclosing five plant fetishes. Two identical shel, which are made of a simple folded ribbon of palmyra palm leaf, usually flank the divination stone to assist with the smooth progress of the divination. Shel has a soothing effect and removes possible dangers from the world of the invisible, especially wandering souls. It would have been passed on to the first diviner by the dye-roon (village chief) who himself would have received it directly from the bush spirits. The other objects have the role of channeling the capricious nature of the bush spirit, which can indeed sulk and "tell lies" as soon as it thinks that the diviner is failing to perform his duties towards it. Kool la kpa, the half-pod of the kpaal fruit (Supplementary Video: 1 min 33 s), is used by bush spirits as a cup for drinking. Just like a threat commonly uttered against humans over their calabashes, the words said over this object expose the bush spirits to stomach aches if they tell lies. The dwo fan fetish, a small piece of the petiole of a palmyra palm leaf, suggests the male section of the dwo cult, and can make a wrongdoer bloat "to the size of a house." Tutya, a tiny bundle of grasses used for roofing and tied in three places, suggests the feminine section of the dwo cult and exposes liars to having their "fingers cut off" (leprosy), or to the total blockage of their urinary and intestinal excretions. These threats are also directed at the diviner.

The Matrimonial Metaphor by Plants

Settling a bush spirit in the diviner's home is metaphorically described as a marriage.

Several objects made of plants that are used in the rites allude to the three stages of the marriage, which the Sèmè assimilate with the initiation of women to the dwo cult.

The little winnowing basket hung under the altar of the female bush spirit is an explicit reference to the one in which adolescents receive gifts from their family and friends during their rites of passage. The woman who holds it also alludes to the woman who provides and carries the kukwan basket in which are placed the gifts for a girl during the first stage (tyèl) of her marriage.

The palmyra palm tree, the leaves of which are used to weave the winnowing basket and the basket, is cultivated in the Orodara region because it is a little further North of its natural area of distribution. Its economic value is important (fruits, sap for wine, stipes for building, leaves for basketry). Its symbolic value is also very strong, suggesting both the ancestors and initiation.

Palmyra palm trees belong to the heads of families and thus ritual objects made with its leaves are powerful and some are dedicated to the dwo cult.

The raw food (fel) offered to the bush spirit suggests the fyè, a small quantity of food presented to the "luck" of the wife during the second stage of marriage (fyè gbe). The "fetish of luck" is placed in the courtyard of their fathers for men, in that of their husbands for women. The meal prepared by the woman chosen by divination is served only to the bush spirit and its followers, just as the one prepared with fyè is only for initiates during a wedding.

The Sèmè do not remember the story of the arrival of fel and they present this plant as inedible and of foreign origin. Some elders cultivate it at home to have it at hand when needed, a common practice for rare, useful plants. Fel is the Lima bean which, according to Tourte (no date), was among the first introductions of American plants on the West African coast. It apparently spread throughout the interior of the continent, but especially in the northern fringes of the forest because, in the savannah, the cowpea, which is well adapted to the environment and meets the needs of the people perfectly, continues to dominate. The plant seems to have been adopted the Sèmè only for mystical uses (e.g. protection against witches). The Gurunsi of Burkina Faso, who also belong to the Volta River Basin cultural area, use it for the same purpose (Saïbou Nignan, pers. comm., November 2016).

The "divination tail" has been equated by elders to a traditional lamp used during the third and final stage of marriage (tyel-de). This is because both have power against witches and are given to the man who assumes responsibility for the marriage.

Discussion

The conjunction of the visible and invisible worlds in the house of the bush spirit is a crucial expected effect of the rites introducing a bush spirit into a human home. As transpositions of objects used daily by humans, some of the plant parts used in the rites illustrates this conjunction. They allude to the lives of bush spirits in the bush: the latters' houses are trees, their ropes are lianas, they drink the water of rivers in wild tree pods, etc. For the Sèmè, the bush is a metaphor of the bush spirits invisible world. In the representations of the inhabitants of the cultural area of the Volta River Basin, the village and the bush are strongly contrasted, the village being the domain of humans and social activities, while the bush is that of invisible beings and the place of all dangers. However, the bush is also a place of nourishment and economic value,

where fields are open and many wild resources are collected for food, crafts, medical care and so on. The languages of the societies of the West African savannah stress this by using the same word for the natural vegetation and the fields. In the same way, bush spirits have two faces: they have certainly instilled in humans all their practical and spiritual knowledge, but punish them when they do not comply with the rules and sometimes take pleasure in misleading their steps or their minds. It is understandable therefore, that transferring a bush spirit from the bush to the house of a diviner is a particularly delicate operation.

The introduction of a bush spirit into the human world mimics the gradual stages of the Sèmè marital union. Such a marital metaphor expressing the individual link established between a human and a supernatural entity is widespread throughout Africa, but it is extended by the Sèmè in unusual detail. Several objects made with plants (baskets, "divination tail"), as well as the plant food of the bush spirit allude to marriage. The effigy of the bush spirit is a token of the transformation of the bush spirit into a domesticated being able to live with humans and like them in a house. Different methods may be adopted to build it in the societies of the Volta River Basin cultural area. Among the Sèmè it is simply made of the wood of a specific tree species, while among the Bwaba, it is made of earth (Dugast 2015a).

In their reading of various ethnographic aspects in other societies of the Volta River Basin, some authors [START_REF] Cartry | Du village à la brousse ou le retour de la question[END_REF][START_REF] Bonnet | La procréation, la femme et le génie (les Mossi de Haute-Volta)[END_REF][START_REF] Hamberger | Traces des génies[END_REF] have concluded that the bush spirit transports its bush world with it. Among the Sèmè, the rites indeed transform the bush spirit's house into a singular and permanent enclave where the world of the village and that of the bush become compatible with each other. Furthermore, the symbolic link thus established between the bush and the house of the bush spirit also extends to the divination scene, wherever the latter may be located. A plant object fulfills this role: a small root readily visible on the divination scene, which is supposed to "sum up" the ritual objects kept in the bush spirit's house. Some other societies in the Volta River Basin have similar ways of materializing with plants the place of origin of the power at work in divination. One option is topological (wood from a particular sacred place), another is botanical (wood from a particular tree species). Among the Kasena the option is topological and the divination wand is carved from a tree growing in one of the sacred groves that are anchor points of social groups; the oracular power is supposed to originate from this grove (Liberski-Bagnoud 2010a). Among the Bwaba (Dugast 2015a), the option is botanical: the earth effigy of the bush spirit always includes an inner frame of wood that ensures its solidity. In the region of Bondoukuy (unpublished surveys related to this project), it is chosen from a closed list of five species by means of a divination sacrifice of chickens. This would be the species of the tree under which the spirit lived but the place is not specified. The Sèmè combine the topological and botanical options. Their wooden effigies of bush spirits, made of specific plant species, in addition symbolically connect the diviner to well-defined natural sacred sites in the bush, which are the anchor points to the territory of the sub-districts or the extended families that make up the clans of Sèmè society [START_REF] Fournier | Setting up the first components of the person and its anchoring to the territory among the Seme of Burkina Faso: "services rendered by ecosystems[END_REF].

The Sèmè bush spirits are thought to assemble the various constituents of the human to be born and hence to have an intimate knowledge of human beings, which is crucial in divination [START_REF] Fournier | Setting up the first components of the person and its anchoring to the territory among the Seme of Burkina Faso: "services rendered by ecosystems[END_REF]. The idea of bush spirits that shape humans is also found among the Bassar [START_REF] Dugast | Le rite de tigiikaal pour les génies de marigot (Bassar du Togo)[END_REF] 2 while among the Kassena the grove that provides the power of divination is also symbolically the place of origin of human groups and the source of its children (Liberski-Bagnoud 2010a)3 .

In addition, among the Sèmè, some of the plant fetishes handled on or around the stone during divination adhere divination to the cult of the dwo, a supernatural entity supposed to have come from the bush. The dwo institution, which is central in Sèmè society, also exists in similar forms among several other societies in the Volta River Basin [START_REF] Petridis | Buffalo Helmets of Tussian and Siemu Peoples of Burkina Faso. African Arts, UCLA James S[END_REF].

Conclusion

This article demonstrated how the Sèmè initiate their new diviners and how they ritually establish the links that are required by divination between the world of humans and the world of the bush and the supernatural. By using plants, they seek a bush spirit in the bush, build an image of it, establish an alliance between it and the novice diviner it will assist, confirm this alliance, and protect them both. By using plants, they also set up the divination scene and obtain "the truth" during divination. Some of these plants are emblems common to society as a whole, with reference in particular to initiation to the dwo cult and marriage. Others are identity markers of the sacred sites that are territorial anchors origin and of clans or families. The Sèmè think that the bush spirit that live in these sacred sites assemble the components of the persons to be born and guide their personal destiny. Thus, to go into the details of the role of plants in the rituals relating to divination is also to access the representations of the Sèmè and their own way of articulating among themselves the central institutions of their society. The representations underlying the Sèmè practices described are also found in several other societies of the Volta River Basin cultural area. However, the use of plants by the Sèmè in this context is exceptionally abundant.

Other societies in Africa practice "divination by plants" in which the oracle is rendered directly by parts of plants or vegetable matter [START_REF] James | The 'listening ebon' Revisited[END_REF][START_REF] Ba | Du signe au blason : description des robes et des marques distinctives du bétail chez les Peuls Fulaabe de l'est du Sénégal[END_REF]. The Sèmè stand out from those other African societies in that they practice rather a "divination with plants". The plants do not participate directly in the oracular process (with the exception of the divination wand). They are only companions, but essential companions. The short sequences presented are extracts from four divination sessions conducted by four different diviners. They show the routine use of the plant material mentioned in the text. These gestures remain the same whatever the subject of the divination.

Supplementary Material

Figure 1 :

 1 Figure 1: A diviner's materials, Orodara region (November 23, 2014)

Table I .

 I Names of the plants cited: script in the International Phonetic Alphabet

	Landolphia dulcis (Sabine) Pichon	kpéél-tokplaa	kpɛǃɛĺ-tokplaá̄/kpéél, burial/
	Landolphia heudelotii A.DC.	kpéél	kpɛǃɛĺ
	Latin binomials Pericopsis laxiflora (Benth.) Meeuwen	Simplified script International phonetics kukwal ou kpokpal kuḱwaĺ, kpoḱpaĺ	Comments
	Phaseolus lunatus var utilis L.	fel	fel
	Abrus precatorius L. Pteleopsis suberosa Engl. & Diels	shunshuun tyaal	ʃùnʃuùn tyaǃaĺ
	Andropogon chinensis (Nees) Merr.	gniin	ɲiǃiń
	Andropogon pseudapricus Stapf Saba senegalensis (A.DC.) Pichon	gniin bwoo	ɲiǃiń bwoǃoŚ
	Afzelia africana Sm. ex Pers. chizachyrium sanguineum (Retz.) Alston	kpaal gniin	kpaǃaĺ ɲiǃiń
	Borassus aethiopum Mart. Zanthoxylum zanthoxyloides (Lam.) Zepern. &	kpeén wo	kpeēń wó
	Cassytha filiformis L. Timler	baarn-byel	baāŕn-byɛl	/hare, possessive, porridge/
	Dichrostachys cinerea (L.) Wight & Arn	blibli	blibliÈ
	ragrostis tremula (Lam.) Hochst. ex Steud	janjur	jaǹjur
	Dioscorea abyssinica Hochst. ex Kunth	kursana	kursańaD
	iospyros mespiliformis Hochst. ex A.DC.	komo	kɔmɔĒ
	ragrostis tremula (Lam.) Hochst. ex Steud	djanjur	jaǹjur	broom
	Gardenia aqualla Stapf & Hutch.	tyéén-kaar tiin	tyɛ̰ ǃɛ̰ ń-kaǃaŕ tḭ ḭ́ ̄/genie, ring ?, male/
	" "	tyéén-kaar bobo	tyɛ̰ ǃɛ̰ ń-kaǃaŕ bobó/ genie, ring ?, deaf-mute/
	Gardenia erubescens Stapf & Hutch.	tyéén-kaar gnéén	tyɛǃɛń-kaǃaŕ ɲɛǃɛń*	/genie, ring ?,female/
	Gardenia ternifolia Schumach. & Thonn.	tyèèn-kaar tiin	tyɛǃɛń-kaǃaŕ tḭ ḭ́ *	/genie, ring ?, male/
	Khaya senegalensis (Desr.) A. Juss.	téen	téǃén	/mahogany tree

Table II :

 II Other terms in Roman and the International Phonetic Alphabet dwo, dolo, pour-put/ literally, pour the dolo of the dwo, the initiation stage for men in the dwo cult dye-roon dye-rɔǃɔń the village chief, authority with a mainly political role fyè fyɛ̰ a small quantity of food used ritually during the second stage of marriage fyè gbe fyɛ̰ gbē/ fyè to place/ literally "to place the fyè ", that is, a part of the food, the second stage of marriage ngmil ta kool ŋmil̄ta koól̀A personal genie that guides the soul and intervenes in the birth of every new human being gna-fléé ɲa-fǃɛǃɛ́/ eye to sprout or to split/ literally, to pierce the eyes, open the eyes), the initiation ancestor, to place/, literally, place the ancestor, a ceremony where a stone is placed on the altar of the ancestors kool gniné mon koól ɲińɛ-mɔń /genie, to sit, thing/ literally, a thing for the genie to sit on) the seat of the genie kool kpé kool kpé "Piquer le genie" "stake down the genie" kool la kpa koól lākpá/ genie, to drink, calabash/ literally, the calabash for the genie to drink from kool naan diir koól naāń diīr the roots of the genie (/genie, wood roots/), placed in the water in small pots kool naan dirn koól naāń dirn the root of the genie (/genie, wood, root/), forked root, accessory for divination kukwan kukwaǹ a basket made of palmyra palm used during the tyèl mukaal muḱaál the name of a sort of porridge usually eaten by the Sèmè sé-daar sɛ̰ -dǃaǃaŕ /divination tail/ literally, tail for divination, a portable "fetish" containing plants sé-daar bar mon sɛ̰ -dǃaǃaŕ bar mɔń /divination, tail, to plug [or wrap around], thing/ literally, the thing that plugs the divination tail), collective name for plants used to make the sé-daar fetish" made of niin grass that forms part of the materials for divination tyéén kan tyɛ̰ ǃɛ̰ ń ka̰ ǹ /genie leg/ literally, the leg of the genie) wooden altar of the genie tyéén no lo tyɛǃɛń nɔ̄lǃō( /genie, mother, show/ literally, honor the mother of the genie) second ritual to welcome the genie tyèl tyɛl the first stage of initiation for adolescents that once included excision and circumcision tyèl-dé tyɛl-dé/ marriage to enter/ literally, to enter into a marriage, third and last stage of the marriage and the initiation for women

	Roman script a bré hlo gnaal taa a ta kukuur a tobra a yen bètaar biin bonoo bunkar dèe donoblè dwo dwo-fan dwo-fwoo kaar-ja mon Téenton kar kar-bo shel sèmè jèèn dwotye gbe gnaan-toon gnyèèl kar ja Jonosoo kaar kono-gnée kɔńǃɔ-ɲɛɛ̄/ sé-ti International phonetic script Translation a bréhlɔ̄ɲaál taā́/ it sets truth road on/ literally, it puts one on the right path a ta kukuúr /it, on, to clean or to delete/ literally, it cleans (the divination stone) a tobra it summarizes, to summarize (to gather) a yɛń it summarizes (to show) bətaár bḭ ḭ̄ ń /winnowing basket little/literally, little winnowing basket bɔńɔɔ̀a personal fetish of luck possessed by everyone buǹkar (/in a group, kar/) a ring of plant material placed inside the point of the roof deè a spouse, a husband dɔńɔblèt he initiation stage for men in the dwo cult dwɔ-fan a little fetish made of a palmyra palm stipe that forms part of the divination materials kaǃaŕ-jàmɔń /creeper, grasp, thing/ literally, thing for the frame) a ring of interlaced plant Teéntɔn "Under-the-mahogany", the name of a district dwɔ-fwǃɔǃɔ́o ne of the sacred groves of the dwo kar kar-bɔ̄c luster of lianas perhaps the name of a species (unidentified) a little fetish made of a knot of a palmyra palm leaf ʃel roof frame sɛmɛ̀jɛɛn the Sèmè language amulet made with plants, a ring made of plant material that is placed around a dwɔ́t he initiation cult shared by several West African societies ɲaǃań-tɔǃɔń the earth chief, an authority with a mainly religious role ɲɛɛĺ kar /rope to cut/ literally, to cut the rope, a ritual to end the period of mourning for widows jà to grasp Jɔǹɔsɔɔ God (the creator) kaāŕ scorpion material placed inside a roof frame sɛ̰ -tí/ divination, stick/, divination wand to tɔ̄f ather dwɔtyɛ gbē/ stage of the dwo kle klɛ̄t o do tutya tutyǃa̰ a little "

See alsoMichel Cartry's summary (2010).

However, the supernatural oracular agency does not necessarily spring from the place of origin of the diviner's group(Liberski-Bagnoud, 2010b)

Acknowledgements

Our thanks to Saïbou Nignan for his assistance in identifying species, Gwenaëlle Fabre et Raymond Boyd for access to the linguistic database, Hamadou Coulibaly for translations from Sèmè-jéen and for the identification of the informants, Deborah Taylor for the translation into English, Jean-Pierre Jacob and Klaus Hamberger for their review of preliminary versions and the UMR 208 Paloc at IRD and the RADICEL-K program for financing.


Ségolène Belletante

Ludovic Montastruc
email: ludovic.montastruc@ensiacet.fr

Stéphane Negny

Raphaële Théry Hétreux

Serge Domenech

Heat Integration optimization in a Multiproduct Biorefinery

Keywords: Heat integration, biorefinery, optimization

The biorefining is a fast-growing topic and laboratory data about biorefineries accumulate sharply, but this type of process is still mainly unknown at the industrial scale. In this context, it is necessary to propose a method that permits to evaluate the industrial interest in order to design and to build the biorefinery. Moreover, the optimization of water and energy consumption represents two of the most important operating costs in a biorefinery. Thus, to limit utilities consumption, energy integration has to be incorporated for all process design alternatives. The proposed MILP program minimizes utilities consumption in coupling cold streams and hot streams through heat exchangers.

Introduction

Nowadays, the main raw material for the chemical industry and energy supply is crude oil. As petroleum resources dwindle permanently, it is necessary to define alternative renewable raw materials and the processes to treat and convert them in order to replace crude oil. The biomass, and especially non-food biomass, is a potential substitute to crude oil for several reasons. Firstly, biomass is a renewable feedstock, thus the environmental impact of chemical industry could be reduced to combat global warming and climate change. Secondly, there are multitude potential conversions pathways for biomass to produce high added value products. Thirdly, the quantity of bio-based wastes increases a lot currently, so their conversion could allow creating a circular economy. Biomass as renewable raw material will definitely be on the rise in deciding countries strategy because it has the potential to contribute to fill chemical and energy needs to ensure countries independence. Currently, to valorize biomass wastes many processes like pyrolysis, gasification, liquefaction and biorefining exist. In this paper, we will focus on the process of biorefining. The biorefining is a fast-growing topic and laboratory data about biorefineries accumulate sharply, but this type of process is still mainly unknown at the industrial scale. Therefore, the economic and ecological viability of the biorefineries in the current level of development is uncertain, especially compared to a perfectly optimized process like the traditional refinery. In this context, it is necessary to propose a method that permits to evaluate the industrial interest in order to design and to build the biorefinery. Moreover, this method should include the design and the optimization of the process in order to make attractive the biorefinery compared to petrochemical process. As a result, the first contribution of this article is to propose a multiscale methodology for the optimal design of multi product biorefinery.

In this methodology, one crucial step concerns heat integration due to large water flowrate and large utilities consumption. Some previous researches focused on the optimization of water and energy consumption, because they represent two of the most important operating costs in a biorefinery. Especially, (Ahmetovic el al. 2010;Grossmann and Martin, 2010;Cucek et al., 2011) worked on the minimization of energy and water consumption in first and second generation biorefinery producing bioethanol. They proposed a two-steps model which minimalized energy consumption by designing the biorefinery from a superstructure with a MINLP program, and then optimized water network to minimize freshwater. They propose a very complete superstructure which encompassed the new technological breakthroughs in the domain. In these previous works it is possible to reduce with some assumptions the number of streams to consider in the heat integration in order to use a classical MINLP program, but in this study no such reduction is stated. Indeed in biorefineries, the heat integration problem contains several cold and hot streams; as a result it is necessary to define a new strategy to solve this problem. In our approach, we propose to use MILP program coupled to scenarii. The program minimizes utilities consumption in coupling cold and hot streams through heat exchangers. The different scenarii vary according the maximal number of heat exchangers allowed for each streams and the approach temperature. A post optimal analysis coupling with a multi criteria decision analysis is used to select the best compromise in term of heat integration.

Methodology

Multiscale modelling

In this section, the methodology is explained in details. The most important point of the proposed model is the multi-scale integration [START_REF] Belletante | Optimal design of an efficient, profitable and sustainable biorefinery producing acetone, butanol and ethanol: Influence of the in-situ separation on the purification structure[END_REF]: it grows from molecule scale to biorefinery scale, including operating unit and process scales, and each level interacts with others iteratively. The thermodynamic and microbiological data at the molecule scale are used to describe the different unit operations composing the different process phases. Then, during the calculation, we use in turn molecule, unit operation, process and biorefinery scales in function on required data. This integration is depicted in the Figure 1. Moreover, the model can be used in any direction, for instance the starting point of the model can be a desired product or a specific feedstock. We remark that the unit operation scale is very important in the model because it represents the interaction between constrains of the process scale and the experimental data extracted from the molecule scale. Furthermore, the unit operation scale will serve to calculate accurate results based on rigorous equation instead of stating hypothesis. In order to explain the Figure 1, we propose to follow the same way described in the methodology: 1a. Choose the potential feedstock and determine its composition. 1b. Define the required bioproducts and physicochemical properties related on. 2. Take an inventory of any microorganism producing the required bioproducts. 3. Collect data related on the fermentation. 4. Collect thermodynamics data for the separation phase. 5. Collect thermodynamics data for the purification phase. 6. Determine chemical compounds to transform in the pretreatment. 7. Define reactants for the fermentation. 8. Take an inventory of any potential pretreatment transforming the feedstock into the reactants for the fermentation. 9. Optimization with economic, environmental and energetic objectives.

Resolution organigram

To enable the implementation of this multi-scale methodology, a six steps method was created to obtain the optimal process. The method deals first with the determination of target bioproducts. Then, a superstructure is created, based on the knowledge of the user or expert, and finally it is modeled. Energy integration is then added. Economic and environmental assessments are then carried out in order to compare the different alternatives and select the optimal process. However, this step sequence corresponds to the spine of the method. In reality, iterations and loops are needed. A complete iterative scheme of the methodology is proposed in Figure 2 with four loops. The general loop is used to perform the different previous step for all of the process alternatives, while the three other are used to remove alternatives based on different criteria. First, a process alternative is chosen to enter in the resolution workflow. The alternative then passes the first test for determining if the technology is mature enough to have enough knowledge to be able to consider it. For example, the amount of knowledge must be sufficient to be able to model the process accurately. Using the methodology described in a previous work [START_REF] Belletante | Optimal design of an efficient, profitable and sustainable biorefinery producing acetone, butanol and ethanol: Influence of the in-situ separation on the purification structure[END_REF], it is mandatory to simulate the process precisely. For this reason, it is necessary to know the different parameters. If this is not the case, the alternative is removed from the superstructure. This allows you to restrict the superstructure. A second test is then focused on the effectiveness of the alternative according to the literature and calculations information preliminary type 'black box '. Therefore, the alternative is deleted if its effectiveness is low. Finally, the number of alternatives decreased and all alternatives in the superstructure can be modeled and simulated. The last test aims to remove no feasible alternatives after the simulation step, which allows furthering restricting the superstructure. Then, energy integration and assessments are then conducted on these processes to allow their comparison to eventually move towards a unique process.

Data Flow Material Flow

Energy integration

Environmental study

Economical study

Biorefinery

Separation

Energy integration

A biorefinery needs a large amount of energy to operate, especially during the pretreatment and purification phase, due to the huge water flowrate in the process. Indeed, in a biorefinery, we estimate that there are 100 to 500 times more (in mass) of water compared to the production of desired bioproduct. Therefore, energy and utilities consumptions are very substantial, and the heat integration is very important in the biorefinery design optimization. .. The synthesis of the Heat Exchanger Network is performed using a MILP program.. The main constraints concern the maximal number of heat exchanges allowed for each stream and the minimum temperature difference between them because they greatly influence the number of heat exchangers in the process on the one hand and the computational time on the other hand. The more the flows are divided, the more the heat exchangers network is complex and better is the minimization of utilities. However, the heat exchangers added for the energy integration increases also the investment of the biorefinery. In the same way, lower is the minimum temperature difference; lower is the number of heat exchangers. Therefore, we have to impose a limit for the number of heat exchanges and a minimum temperature difference.

Case study for heat integration

The case study is the Acetone-Butanol-Ethanol (ABE) production from lignocellulosic feedstock, especially wood. This paper exemplifies the methodology and the multiscale integration from the pretreatment to the purification phase of the process. Biobutanol is promising product as it can replace ethanol in fuels to improve its efficiency. It is a chemical intermediate to create products with higher added value like others chemical products or plastics. The goal of integration is to reduce consumption in the factory utilities, which allows to improve both the economic and environmental aspects of the biorefinery. Indeed, the reduction of the energy consumption of the process induced some lower operating costs, and the decrease of environmental impact. The problem is classically a MINLP problem similar to the model of [START_REF] Yee | Simultaneous optimization models for heat integration-II. Heat exchanger network synthesis[END_REF]. However, the resolutions are possible for "small" problems that is, for a number of hot and cold streams less than 5. However, in this study, a biorefinery has potentially several tens of hot and cold streams. For example in our case study, the biorefinery contains 32 hot and 31 cold streams. One possible strategy to reduce this number of streams would be to use the methodology developed by [START_REF] Rafione | The green integrated forest biorefinery: An innovative concept for the pulp and paper mills[END_REF]. This methodology determined the "main" streams, namely those which have very significant temperature differences and those which have a large enthalpy. The selection is operated using a multi criteria decision aided tool. For the criteria the enthalpy value and the temperature level absolute value are considered. Using Topsis methodology, a ranking is made for the cold and hot streams. In order to use MINLP, a limiting number (5-6) hot and cold are chosen. However, as the biorefinery contains many identical unit operation characterized by very high enthalpy, it is impossible to implement such a methodology. As a consequence, the proposed strategy is to turn the problem into a MILP formulation, i.e. to no longer calculate the exchange area and the mean logarithmic temperature difference. Thus, different scenarios are described by setting the number of double heat exchangers, the minimum temperature difference and the number of possible divisions by stream. For each scenario, the optimization criterion is the minimization of the sum of hot and cold utilities consumed. Scenarios ( Tmin, number of division and number maximum interchange of integration) are compared by calculating the Net Present Value and the environmental impact. For the energy integration, we studied the influence on the profit of the minimum temperature difference (5 and 10°C), of the maximal number of heat exchanges for each stream (one or two) and the number of integration heat exchangers (between 10 and 30). The profit is calculated in using the investment of heat exchanger network and the utilities cost. The energy integration is made on the entire process. According to the table 1, regardless the scenario, the energy integration allows making profit of 20% at least. The scenario with one division and a minimum temperature difference of 10°C is abandoned because it presents the lowest profits while it offers lowest investments. That is the heat exchanger network is less complex than for scenarios with 2 divisions or with a minimum temperature difference smaller; but in turn, the utilities consumption is higher. The scenarios with a minimum temperature difference of 5°C seem more interesting, especially the two scenarios with 25 or 30 integration exchangers (grey in the table 1).

Conclusion

In support of a framework for biorefinery synthesis design, we have presented a generic methodology to optimize a biorefinery with a rigorous thermodynamics approach. The multiobjective-optimization problem maximizes the profit and minimizes the environmental impact. A biorefinery needs a large amount of energy to operate. Thus, to limit utilities consumption, energy integration has to be incorporated for all alternatives. The energy integration is implemented as a MILP program in order to perform the economic and environmental impact in the same time.

Figure 1 :

 1 Figure 1: Illustration of the multi-scale modeling and optimization of biorefineries

  Figure 2: Resolution Organigram

Table 1 .

 1 Result for each scenario Number of divisions, 2 N.H.E.: Number of heat exchanger, 3 C.C.: Capital Cost, 4 A.O.P.: Annual Operating Costs, 5 C.B.: Cost Benefit, 6 E.B.: Environmental Benefit

	N.D. 1	T min N.H.E. 2 C.C 3 (M$) A.O.C. 4 (M$/year) C.B. 5 (%)	E.B 6 (%)
	0	0	0	11	60	0	0
	1	5	25	21.3	44.1	62.5	30.4
	2	5	25	23.6	39.2	82.5	38.2
	1	10	24	20.5	46.9	50.6	25.9
	1	5	15	23.4	40.7	75.9	36.1
	1	5	10	21.1	45.2	57.7	31
	1	5	20	23.5	39.7	80.3	38.8
	1	5	28	20.9	46.2	53.4	25.6
	2	5	30	23.8	39.1	82.8	38.4
	1	5	20	20.8	44.3	62.0	30.1
	1	5	15	18.8	44.8	61.5	29.7
	1	5	10	20.2	45.6	56.7	28.1
	1	10	10	17.5	48.3	46.9	24.2
	1	10	15	18.3	47.5	49.8	25.4
	1	10	20	19.4	47.1	50.7	25.7

1 N.D.: