Víctor Alcaraz-González
email: victor.ga@redudg.udg.mx

Azael Fabián

Fregoso-Sánchez

Víctor Alvarez-González

Jean-Philippe Steyer
email: jean-philippe.steyer@inrae.fr

Alcaraz -González

V ; Fregoso-Sánchez

Jérôme Harmand

Alain Rapaport

Neli Dimitrova

Ivan Simeonov

Multivariable Robust Regulation of Alkalinities in Continuous Anaerobic Digestion Processes: Experimental Validation

Keywords: anaerobic digestion, alkalinity, control, multivariable, uncertainties, robustness, wastewater treatment

A multivariable adaptive feedback control for highly uncertain continuous anaerobic digestion processes is proposed to regulate the volatile fatty acids (VFA) concentration, the strong ions concentrations, and the total and intermediate alkalinities. The multivariable control scheme includes a Luenberger observer to estimate both the unmeasured variables (i.e., VFA) and unknown microbial growth kinetics. The control approach is designed using an exponential Lyapunov function to resemble the typical exponential biological growth of the involved microbial consortia. Taking into account physicochemical equilibrium, alkalinities are represented as a function of the state variables. As a result, the control problem becomes a regulation problem on alkalinities, and in turn, a tracking control problem on the state variables, with two manipulated variables-the dilution rate and the feed rate of a strong alkali solution-while the state variables' set-points are given as a function of pH. The implementation of this multivariable control scheme was experimentally tested and validated in a 0.982 m 3 pilot plant treating agro-industrial wastewater, and demonstrated to be robust in the face of unknown microbial growth kinetics. Results showed the potential for practical application and optimization of industrial digesters.

Introduction

Anaerobic digestion (AD) is a very complex biological process used for treating agroindustrial wastewaters with a high variety of organic long-chain compounds, such as carbohydrates, lipids, and proteins. Expressed as the chemical oxygen demand (COD), these compounds represent the substrate for a consortium composed of a large variety of microorganisms that decomposes the organic matter in the absence of oxygen into microbial biomass, residual organic matter, and biogas (mainly composed of methane and carbon dioxide). AD is one of the most widely used bioprocesses for treating agro-industrial effluents [START_REF] Mata-Alvarez | Anaerobic digestion of organic solid wastes. An overview of research achievements and perspectives[END_REF][START_REF] Metcalf | Wastewater Engineering: Treatment, Disposal, and Reuse[END_REF][START_REF] Mata-Alvarez | A critical review on anaerobic co-digestion achievements between 2010 and 2013[END_REF][START_REF] Rehman | Anaerobic digestion[END_REF], capable of directly treating high organic load wastewater (10-80 gCOD L -1 ) [START_REF] Steyer | Evaluation of a four year experience with a fully instrumented anaerobic digestion process[END_REF][START_REF] Zhang | Characterization of food waste as feedstock for anaerobic digestion[END_REF][START_REF] Appels | Principles and potential of the anaerobic digestion of waste-activated sludge[END_REF][START_REF] Chan | A review on anaerobic-aerobic treatment of industrial and municipal wastewater[END_REF]. It is relatively easy to implement in practice [START_REF] Angelidaki | Applications of the Anaerobic Digestion Process[END_REF], reaches high removal yields (between 90 and 98% COD) [START_REF] Rehman | Anaerobic digestion[END_REF][START_REF] Mir | Design considerations and operational performance of anaerobic digester: A review[END_REF], and produces an added-value byproduct-biogas-which is rich in methane (60-70% content) [START_REF] Mir | Design considerations and operational performance of anaerobic digester: A review[END_REF] and may be used as fuel or recycled in the grid [START_REF] Appels | Principles and potential of the anaerobic digestion of waste-activated sludge[END_REF][START_REF] Holm-Nielsen | The future of anaerobic digestion and biogas utilization[END_REF]. In contrast, AD processes may present some operating difficulties since the excessive accumulation of volatile fatty acids (VFA) may lead the process to a breakdown. Moreover, it is well known that AD exhibits a highly nonlinear behavior [START_REF] Donoso-Bravo | Model selection, identification and validation in anaerobic digestion: A review[END_REF][START_REF] Bernard | Can we assess the model complexity for a bioprocess: Theory and example of the anaerobic digestion process[END_REF], which makes any control scheme more difficult to design and develop. Furthermore, it has one of the most diverse ecosystems in nature [START_REF] Godon | Microbial 16S rDNA diversity in an anaerobic digester[END_REF][START_REF] Delbã S | Bacterial and archaeal 16S rDNA and 16S rRNA dynamics during an acetate crisis in an anaerobic digestor ecosystem[END_REF], and their growth kinetics are difficult to model [START_REF] Husain | Mathematical models of the kinetics of anaerobic digestion-A selected review[END_REF][START_REF] Gavala | Kinetics and Modeling of Anaerobic Digestion Process[END_REF]. Furthermore, even when AD withstands high organic loads, drastic process input step changes may result in a partial or even complete loss of operational stability (OS) [START_REF] Angelidaki | Effect of operating conditions and reactor configuration on efficiency of full-scale biogas plants[END_REF][START_REF] Méndez-Acosta | A robust control scheme to improve the stability of anaerobic digestion processes[END_REF]. Particularly, regarding this last issue, alkalinity is recognized among the most important variables for preserving the OS in AD processes [START_REF] Ripley | Improved alkalimetric for anaerobic digestion wastes monitoring of municipal sludge[END_REF][START_REF] Moosbrugger | Weak acid/bases and pH control in anaerobic systems-A review[END_REF]. On the other hand, it has been reported that a relatively high presence of strong ions provokes unbalancing of the physicochemical equilibrium and may result in inhibition that directly affects the methanogenic archaea population [START_REF] Chen | Inhibition of anaerobic digestion process: A review[END_REF][START_REF] Chen | Effects of alkalinity sources on the stability of anaerobic digestion from food waste[END_REF] and eventually the bioprocess acidification [START_REF] Jarrell | Inhibition of methanogenesis in pure cultures by ammonia, fatty acids, and heavy metals, and protection against heavy metal toxicity by sewage sludge[END_REF][START_REF] Fagbohungbe | The challenges of anaerobic digestion and the role of biochar in optimizing anaerobic digestion[END_REF][START_REF] Buyukkamaci | Volatile fatty acid formation in an anaerobic hybrid reactor[END_REF][START_REF] Hess | Design and study of a risk management criterion for an unstable anaerobic wastewater treatment process[END_REF]. When this takes place, the system is wrapped in a vicious circle: higher acidification, greater inhibition, and greater inhibition, greater acidification; thus, the system collapses. Thereby, the main inhibition indicator, and thus also of OS, is alkalinity, because it involves all the physicochemical system equilibria [START_REF] Liu | Equilibrium Model of Anaerobic Reactors[END_REF][START_REF] Florencio | pH-Stability in anaerobic bioreactors treating methanolic wastewaters[END_REF][START_REF] Lahav | Titration methodologies for monitoring of anaerobic digestion in developing countries?a review[END_REF].

Only a few studies have been developed for controlling alkalinity in the past. For instance, Wang et al. [START_REF] Wang | Evaluation of artificial neural network models for online monitoring of alkalinity in anaerobic co-digestion system[END_REF] developed an artificial neural network to monitor alkalinity whereas Barampouti et al. [START_REF] Barampouti | Dynamic Modeling of the Ratio Volatile Fatty Acids/Bicarbonate Alkalinity in a UASB Reactor for Potato Processing Wastewater Treatment[END_REF] used a lagged regression model to describe the VFA/bicarbonate ratio's alkalinity behavior in AD processes. These studies focused on predicting the operating conditions to indirectly prevent the process breakdown. On the other hand, other control approaches also have been conducted to guarantee the operational stability by regulating the pH or by controlling both the VFA concentration and pH [START_REF] Méndez-Acosta | A robust control scheme to improve the stability of anaerobic digestion processes[END_REF][START_REF] Méndez-Acosta | Robust Control of Volatile Fatty Acids in Anaerobic Digestion Processes[END_REF][START_REF] Garcia-Sandoval | VFA robust control of an anaerobic digestion pilot plant: Experimental implementation[END_REF]. However, to the knowledge of the authors, regulating the alkalinity directly has not been fully addressed. The reason for this is simple: only a few dynamical AD models describe alkalinity as a state variable because it is a function of pH. Thus, its derivative with respect to time, though possible, would result in expressions difficult to handle and allocate in control law calculations. In order to facilitate the understanding of the AD process, the International Water Association (IWA) [START_REF] Batstone | The IWA Anaerobic Digestion Model No 1 (ADM1)[END_REF] has developed an AD model, the so-called ADM1, which has been extensively used for simulation purposes, state variables monitoring and estimation [START_REF] Montiel-Escobar | ADM1-Based Robust Interval Observer for Anaerobic Digestion Processes[END_REF], experimental validation and optimization [START_REF] Torres Zúñiga | Experimental validation of online monitoring and optimization strategies applied to a biohydrogen production dark fermenter[END_REF], and even for validating simpler models [START_REF] Giovannini | On the derivation of a simple dynamic model of anaerobic digestion including the evolution of hydrogen[END_REF]. The well-known ADM1 model takes into account the cations and anions activities that resulted either in differential equations (DE) or differential algebraic equations (DAE) sets [START_REF] Batstone | The IWA Anaerobic Digestion Model No 1 (ADM1)[END_REF][START_REF] Patón | Activity corrections are required for accurate anaerobic digestion modelling[END_REF]; nevertheless, its pH functionality is not explicit. Recent efforts have been addressed to deal with this important operating variable. For instance, Zhang et al. [START_REF] Zhang | Improved ADM1 model for anaerobic digestion process considering physico-chemical reactions[END_REF] modified the ADM1 model by integrating a more detailed physicochemical framework and by introducing inorganic carbon and nitrogen balance terms to resolve the discrepancies between the carbon and nitrogen contents, while Shi et al. [START_REF] Shi | Modeling the dynamic volatile fatty acids profiles with pH and hydraulic retention time in an anaerobic baffled reactor during the startup period[END_REF] improved the ADM1 model using a variable stoichiometric approach based on the high hydrogen partial pressures and pH. Although these improvements provide useful information, the high dimension of the modified ADM1 makes it very difficult to implement in process control structures [START_REF] Xue | Nonlinear model predictive control of anaerobic digestion process based on reduced ADM1[END_REF]. In this sense, efforts have been made to reduce the ADM1 complexity, which under certain practical and theoretical considerations yields more manageable AD models for control purposes [START_REF] Xue | Nonlinear model predictive control of anaerobic digestion process based on reduced ADM1[END_REF][START_REF] Hassam | A generic and systematic procedure to derive a simplified model from the anaerobic digestion model No. 1 (ADM1)[END_REF][START_REF] Kiely | Physical and mathematical modelling of anaerobic digestion of organic wastes[END_REF]. One of these reduced models was developed almost at the same time than the ADM1 model; it is the so-called AM2 model [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF], whose structure does not explicitly contain the alkalinity as a function of pH, but does include a state variable named Z, which takes into account the concentration of all the strong ions (cations and anions) in the bulk phase, and considers that, at a pH close to 7, it is proportional to total alkalinity [START_REF] Hassam | A generic and systematic procedure to derive a simplified model from the anaerobic digestion model No. 1 (ADM1)[END_REF]. By performing the corresponding mass balances, Z, the VFA concentration, and the total inorganic carbon concentration become state variables in the AM2 model. It has been successfully experimentally validated on a 0.982 m 3 fixed-bed bioreactor pilot plant used for the treatment of red wine vinasses [START_REF] Steyer | Evaluation of a four year experience with a fully instrumented anaerobic digestion process[END_REF]; in fact, the same pilot plant is used in the present work. Indeed, this model has been very useful for describing AD processes, treating mainly soluble, carbohydrate-based organic matter and where degradation of complex and proteinaceous substrates, such as waste-activated sludge, is negligible [START_REF] Hassam | A generic and systematic procedure to derive a simplified model from the anaerobic digestion model No. 1 (ADM1)[END_REF]. In addition, it has been largely used in optimization, parameter and state estimation, and process control approaches in AD (for instance, see [START_REF] Theilliol | On-line estimation of unmeasured inputs for anaerobic wastewater treatment processes[END_REF][START_REF] Didi | A Luenberger-type observer for the AM2 model[END_REF][START_REF] Rodriguez-Liñan | An adaptive observer for operation monitoring of anaerobic digestion wastewater treatment[END_REF][START_REF] Vargas | A fast extremum-seeking approach for the methanisation of organic waste in an anaerobic bioreactor[END_REF]), as well as in actual instrumentation and control applications [START_REF] Alcaraz-González | Robust interval-based regulation for anaerobic digestion processes[END_REF][START_REF] Bernard | An integrated system to remote monitor and control anaerobic wastewater treatment plants through the internet[END_REF][START_REF] Steyer | Lessons learnt from 15 years of ICA in anaerobic digesters[END_REF][START_REF] Jimenez | Instrumentation and control of anaerobic digestion processes: A review and some research challenges[END_REF]. This model also has been used for monitoring the ratio VFA/Z related to the OS in AD, taking into account alkalinity [START_REF] Attar | Model-based optimal recovery of methane production in an anaerobic digestion reactor. Model. Identif[END_REF] in continuous AD reactors, and for modeling the alkalinity spatial distribution in an up-flow fixed-bed anaerobic digester [START_REF] Hmissi | Evaluation of alkalinity spatial distribution in an up-flow fixed bed anaerobic digester[END_REF]. The ratio VFA/Z also has been regulated by using a model predictive control (MPC) approach in order to achieve an optimal OS on the basis of a modified AM2 model [START_REF] Kil | Model predictive control with on-line model identification for anaerobic digestion processes[END_REF].

The aim in this contribution is to design, develop, and implement a multivariable adaptive feedback controller to regulate the VFA concentration and total and intermediate alkalinities to guarantee the operational stability of the AD process. The controller is a robust multiple-input, multiple-output (MIMO) control, which has been previously described and demonstrated in [START_REF] Alcaraz-González | Robust Regulation of Alkalinity in Highly Uncertain Continuous Anaerobic Digestion Processes[END_REF] by using a modified AM2 model and an interval observer for handling process inputs uncertainties. As in [START_REF] Alcaraz-González | Robust Regulation of Alkalinity in Highly Uncertain Continuous Anaerobic Digestion Processes[END_REF], alkalinity references are translated into state variable references, taking into account pH variations. The proposed control scheme is experimentally tested and validated in a pilot plant anaerobic digester, consisting of a 0.982 m 3 up-flow fixed-bed bioreactor, located at LBE-INRAE, Narbonne, France. The structure of the control schemes takes into account the high nonlinearity of the AD process, the unknown microbial growth kinetics, and its parameter uncertainty [START_REF] Alcaraz-González | Robust Regulation of Alkalinity in Highly Uncertain Continuous Anaerobic Digestion Processes[END_REF][START_REF] Alcaraz-Gonzalez | Software sensors for uncertain wastewater treatment processes: A new approach based on interval observers[END_REF]. A Luenberger observer [START_REF] Bastin | On-Line Estimation and Adaptive Control of Bioreactors[END_REF] is used in the control scheme to estimate the biomass concentrations (which cannot be measured), together with the kinetics and yield coefficients. It is shown that the proposed robust MIMO control scheme is robust in the face of parameter uncertainty, unknown kinetics, and pH variations, as well as in measuring disturbances.

Materials and Methods

Anaerobic Digestion Model

In this work, we used a modified AM2 model, called the AM2a model [START_REF] Alcaraz-González | Robust Regulation of Alkalinity in Highly Uncertain Continuous Anaerobic Digestion Processes[END_REF] in which a second input flow, consisting of concentrated NaOH, is added and is the second manipulated variable. This model is given by: .

X 1 = (µ 1 -αD(t))X 1 . X 2 = (µ 2 -αD(t))X 2 . S 1 = S in 1 -S 1 D(t) -k 1 µ 1 X 1 . S 2 = S in 2 -S 2 D(t) + k 2 µ 1 X 1 -k 3 µ 2 X 2 . Z = Z in 1 -Z D 1 (t) + Z in 2 -Z D 2 (t) (1)
with

µ 1 = µ max1 S 1 k S1 +S 1 µ 2 = µ max2 S 2 k S2 + S 2 + (S 2 /k I ) 2 , and D(t) = D 1 (t) + D 2 (t) ≈ D 1 (t) (2)
where X 1 and X 2 represent the acidogenic and methanogenic biomass concentrations, respectively, S 1 (g L -1 ) is the organic matter concentration expressed as chemical oxygen demand (COD), and

S 2 (mmol L -1 ) is the VFA concentration. D 1 (t) (d -1
) is the main dilution rate and D 2 (t) (d -1 ) corresponds to the concentrated alkali solution dilution rate (here, it is assumed that D 2 (t) <<< D 1 (t). Z (mEq L -1 ) is the strong ions concentration, where Z in 1 (mEq L -1 ) and Z in 2 (mEq L -1 ) are the strong ion concentrations in the main and second input flows, respectively. µ 1 (d -1 ) (Monod type) and µ 2 (d -1 ) (Haldane type) are the microbial specific growth rates for acidogenic bacteria and methanogenic archaea, respectively, where µ max1 and µ max2 (d -1 ) are the maximum growth rates k S1 (g L -1 ), and k S2 (mmol L -1 ) represent the half saturation constants while k I ((mmol L -1 )) 1/2 represents the inhibition constant. Parameters k 1 to k 3 are yield coefficients in the corresponding units. Parameter α (dimensionless) denotes the biomass fraction that is retained for the bioreactor bed; i.e., α = 0 stands for an ideal fixed-bed reactor, while α =1 stands for an ideal continuous stirred reactor tank [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF].

Control Objectives

According to [START_REF] Ripley | Improved alkalimetric for anaerobic digestion wastes monitoring of municipal sludge[END_REF][START_REF] Méndez-Acosta | Robust Control of Volatile Fatty Acids in Anaerobic Digestion Processes[END_REF][START_REF] Alcaraz-González | Robust Regulation of Alkalinity in Highly Uncertain Continuous Anaerobic Digestion Processes[END_REF][START_REF] Alcaraz-Gonzalez | Estimation et Commande Robustes non Linéaires des Procédés Biologiques de Dépollution des Eaux usées: Application a la Digestion Anaérobie[END_REF][START_REF] Drosg | Process Monitoring in Biogas Plants[END_REF], operational stability can be held by keeping the following alkalinity criteria: TA ≥ 3 gCaCO 3 L -1 i.e., 60 mEq L -1

(3)

I A/TA ≤ 0.3 (4)
where I A represents the intermediate alkalinity, and TA represents the total alkalinity. Both I A and TA can be measured by simple titration using the method proposed by Ripley et al. [START_REF] Ripley | Improved alkalimetric for anaerobic digestion wastes monitoring of municipal sludge[END_REF], which consists of bringing the pH in a solution from the initial pH 0 to pH 1 = 5.75 (first point of titration) and then to pH 2 = 4.3 (second point of titration). However, as it was stated above, and as it can be seen in Model (1), neither TA nor I A are state variables. Then, the reference inequalities (3-4) have been translated into true state variables, according to [START_REF] Alcaraz-González | Robust Regulation of Alkalinity in Highly Uncertain Continuous Anaerobic Digestion Processes[END_REF][START_REF] Alcaraz-Gonzalez | Estimation et Commande Robustes non Linéaires des Procédés Biologiques de Dépollution des Eaux usées: Application a la Digestion Anaérobie[END_REF], as follows:

TA ≡ f Tc Z(t) + F a ( f Ta -f Tc )S 2 (t) (5)
I A TA ≡ f Ic Z(t) + F a ( f Ia -f Ic )S 2 (t) f Tc Z(t) + F a ( f Ta -f Tc )S 2 (t) (6)
with

f Tc = 1 -10 -pH(t) +K c 10 -4.3 +K c f Pc = 1 -10 -pH(t) +K c 10 -5.75 +K c f Ta = 1 -10 -pH(t) +K a 10 -4.3 +K a f Pa = 1 -10 -pH(t) +K a 10 -5.75 +K a F a = K a 10 -pH(t) +K a f Ic = f Tc -f Pc f Ia = f Ta -f Pa (7)
Using Equations ( 5)-( 7), the reference for the true state variables S 2 and Z that guarantee the digester operational stability can be obtained as follows:

S r 2 = TA r f Ic -(I A/TA) r f Tc F a (F 1 + F 2 ) (8) Z r = TA r -F a ( f Ta -f Tc )S r 2 f Tc (9)
with

F 1 = f Tc (I A/TA) r ( f Ta -f Tc ) -( f Ia -f Ic ) F 2 = ( f Ta -f Tc ) f Tc -(I A/TA) r f Tc
where S r 2 and Z r are the set-point values for the VFA and strong ions concentrations, respectively, for TA r and (I A/TA) r fulfilling inequalities (3-4). Figures 1 and2 show the explicit pH dependence of S r 2 and Z r , respectively.

of values for the alkalinity criteria 𝑇𝑇𝑇𝑇 𝑟𝑟 , an (𝐼𝐼𝑇𝑇 𝑇𝑇𝑇𝑇 ⁄ ) 𝑟𝑟 below 𝑇𝑇𝑇𝑇 𝑚𝑚𝑖𝑖𝑖𝑖 𝑟𝑟 =60 mEq L -1 , and (𝐼𝐼𝑇𝑇 𝑇𝑇𝑇𝑇 ⁄ ) 𝑚𝑚𝑖𝑖𝑖𝑖 𝑟𝑟 = 0.22 , the AD process must operate at a pH > 8 to satisfy 𝑆𝑆 2 𝑟𝑟 > 0 . However, such a pH is off of the operational pH interval used in AD. In any case, it is clear that while 𝑇𝑇𝑇𝑇 𝑟𝑟 and (𝐼𝐼𝑇𝑇 𝑇𝑇𝑇𝑇 ⁄ ) 𝑟𝑟 are fixed set-points, 𝑆𝑆 2 𝑟𝑟 and 𝑍𝑍 𝑟𝑟 are not, but these are considered as a function of pH. Figure 1 also shows that TA is effectively close to Z (for a pH close to 7), while Figure 2 shows how the alkalinity criterion (4) regarding the ratio I A/TA implies a more careful and restrictive choice of S r 2 , mainly for pH < 7. Actually, the lower the pH, the reference that should be set for S 2 is more restrictive. For instance, for TA r = 60 mEq L -1 , (I A/TA) r = 0.3, and the pH = 6.68, S r 2 is set at zero, as shown in Figure 2, but for a pH = 6.8, and at the same (I A/TA) r = 0.3, S r 2 is set at S r 2 ≈ 2 mmol L -1 . Nevertheless, the minimal possible value for criterion (4) is the ratio I A/TA min = 0.22, which is attainable only in the interval of pH 7.57 < pH < 8, because below these conditions Equation (8) predicts negative S r 2 values. Thus, an important result in this analysis is the fact that for any pair of values for the alkalinity criteria TA r , an (I A/TA) r below TA r min = 60 mEq L -1 , and (I A/TA) r min = 0.22, the AD process must operate at a pH > 8 to satisfy S r 2 > 0. However, such a pH is off of the operational pH interval used in AD. In any case, it is clear that while TA r and (I A/TA) r are fixed set-points, S r 2 and Z r are not, but these are considered as a function of pH.

Control Approach

Once the set-points are established in terms of the true state variables in Model (1), they can be included in the MIMO control laws given by Equations ( 10)- [START_REF] Donoso-Bravo | Model selection, identification and validation in anaerobic digestion: A review[END_REF]. They have the same mathematical structure as the control laws used in the numerical simulations in [START_REF] Alcaraz-González | Robust Regulation of Alkalinity in Highly Uncertain Continuous Anaerobic Digestion Processes[END_REF]. It is worth mentioning that in this work, the control laws were constructed by combining exponential Lyapunov functions and a Luenberger observer to estimate the uncertain kinetics. The resulting MIMO control laws are:

D 1 = -η -λ 1 υ 1 S in 2 -S 2 (10)
D 2 = -D 1 Z in 1 -Z -λ 2 υ 2 Z in 2 -Z (11)
with

υ 1 = 1 -e -1 2 (S 2 -S r 2 ) 2 S 2 -S r 2 ,υ 2 = 1 -e -1 2 (Z-Z r ) 2 |Z -Z r | ( 12
)
and λ 1 and λ 2 being the adaptive parameters that satisfy the inequalities:

0 < λ 1 < max υ 1 , 0 < λ 2 < max υ 2 (13)
Notice that υ 1 , i = {1, 2}, are υ 1 = f (pH). Hence, λ i , i = {1, 2} are actually adaptive parameters that can be suitably chosen by the user, provided that the inequalities [START_REF] Bernard | Can we assess the model complexity for a bioprocess: Theory and example of the anaerobic digestion process[END_REF] 1) represents the uncertain kinetic terms that can be estimated using the following Luenberger observer:

are fulfilled. η = k 2 µ 1 X 1 -k 3 µ 2 X 2 in Model (
. Ŝ2 = η + D S in 2 -Ŝ2 + Γg 1 S 2 -Ŝ2 η = Γ 2 g 2 S 2 -Ŝ2 (14)
where g 1 y g 2 are chosen in such a way that the polynomial s 2 + g 2 s + g 1 = 0 is Hurwitz, and Γ is the observer gain. In this way the Luenberger observer ( 14) guarantees that the state vector S 2 -Ŝ2 , ηη T → ε for t → ε , where ε is a vector of arbitrarily small values around the origin [START_REF] Palacios-Ruiz | Regulation of Volatile Fatty Acids and Total Alkalinity in Anaerobic Digesters[END_REF]. More details about the general control structure design can be found in [START_REF] Alcaraz-González | Robust Regulation of Alkalinity in Highly Uncertain Continuous Anaerobic Digestion Processes[END_REF]. The stability proof for the proposed MIMO control approach ( 10)-( 11), for which -1, with y = {S 2 , Z}, are Lyapunov functions, also can be found in [START_REF] Alcaraz-González | Robust Regulation of Alkalinity in Highly Uncertain Continuous Anaerobic Digestion Processes[END_REF]. Figure 3 is a schematic representation of the proposed MIMO control approach.

V(t) = e

Experimental Setup

Experimental validation was carried out over 12 days on a pilot plant at the French Research Institute for Agriculture, Food and Environment-

Experimental Setup

Experimental validation was carried out over 12 days on a pilot plant at the French Research Institute for Agriculture, Food and Environment-Laboratory of Environmental Biotechnology (LBE-INRAE) facilities, located in Narbonne, France. The experimental AD process is an up-flow fixed-bed bioreactor for the treatment of red wine vinasses, with a useful volume of 0.982 m 3 [START_REF] Steyer | Evaluation of a four year experience with a fully instrumented anaerobic digestion process[END_REF] (see Figure 4). It is fully instrumented with on-line measurements every half hour of the VFA and bicarbonate (Bic) concentrations, as well as TA and I A using a titrimetric sensor, [START_REF] Bouvier | On-line titrimetric sensor for the control of anaerobic digestión processes[END_REF]. Using these measurements, it was also possible to measure Z as Z = VFA + Bic. However, it is important to notice that some technical problems required sensor maintenance in the time periods 5.95 d < t < 6.5 d and 8.9 d < t < 9.6 d. As a consequence, measurements on S 2 and Z as well as TA and I A were not available in these time periods in a proper form (these readings were excessive noise), and as a consequence, they were not used in the calculations and thus excluded in Figures 5678. Another drawback with the NaOH dosage pump, in the time period 9.5 d < t < 10.5 d, prevented Q in2 to follow its respective control law. During this time period, a default Q in2 = 0.028 L h -1 was used with no important consequences in the MIMO controller performance (as we will discuss later in the paper). The average operating temperature was 35 The COD concentration was measured using a mid-infrared spectrometer (Avatar 380 TM FT-IR from Nicolet company modified to remove temperature effects) [START_REF] Steyer | On-line measurements of COD, TOC, VFA, total and partial alkalinity in anaerobic digestion processes using infra-red spectrometry[END_REF]. With the exception of the titrimetric measurements, which were taken every half hour, the rest of the measurements, such as the recirculation liquid flow rate, reactor temperature, and output biogas composition, were available every 2 min, but not all these measurements are shown in this work. Although measurements on the COD and gas flow rate are not involved in the control approach, they are shown and discussed later together with the overall performance of the system. For a complete description of the anaerobic digester process used in this work as well as its instrumentation, the reader is invited to refer to [START_REF] Steyer | Evaluation of a four year experience with a fully instrumented anaerobic digestion process[END_REF], whose information may be also in complemented by [START_REF] Steyer | Lessons learnt from 15 years of ICA in anaerobic digesters[END_REF]. With respect to titrimetric and COD measurements, the reader may obtain supplementary information in [START_REF] Bouvier | On-line titrimetric sensor for the control of anaerobic digestión processes[END_REF] and [START_REF] Steyer | On-line measurements of COD, TOC, VFA, total and partial alkalinity in anaerobic digestion processes using infra-red spectrometry[END_REF], respectively. The experimental run began operating the AD process in open-loop for 12 h to adjust the Luenberger observer parameters, which remained without changes for the whole experimental run. According to the alkalinity criteria (3-4), Table 1 shows the values used as alkalinity references through time, while Table 2 shows the parameter used for the Luenberger observer [START_REF] Godon | Microbial 16S rDNA diversity in an anaerobic digester[END_REF]. All sensor noisy readings were filtered for better plotting purposes. However, it should be noticed that the whole experiment in the 12 days of duration was carried out entirely online, and calculations for both the Luenberger estimates and control inputs were performed using raw online data. The COD concentration was measured using a mid-in spectrometer (Avatar 380 TM FT-IR from Nicolet company modi remove temperature effects) [START_REF] Steyer | On-line measurements of COD, TOC, VFA, total and partial alkalinity in anaerobic digestion processes using infra-red spectrometry[END_REF]. With the exception of the titr measurements, which were taken every half hour, the rest measurements, such as the recirculation liquid flow rate, temperature, and output biogas composition, were available every but not all these measurements are shown in this work. Al measurements on the COD and gas flow rate are not involved in the approach, they are shown and discussed later together with the performance of the system. For a complete description of the an digester process used in this work as well as its instrumentation, the is invited to refer to [START_REF] Steyer | Evaluation of a four year experience with a fully instrumented anaerobic digestion process[END_REF], whose information may be also in complem by [START_REF] Steyer | Lessons learnt from 15 years of ICA in anaerobic digesters[END_REF]. With respect to titrimetric and COD measurements, the read obtain supplementary information in [START_REF] Bouvier | On-line titrimetric sensor for the control of anaerobic digestión processes[END_REF] and [START_REF] Steyer | On-line measurements of COD, TOC, VFA, total and partial alkalinity in anaerobic digestion processes using infra-red spectrometry[END_REF], respectivel experimental run began operating the AD process in open-loop for adjust the Luenberger observer parameters, which remained w Process input concentrations used in the experimental run are shown in Table 3. Notice that the measured S in 2 was held constant at S in 2 = 84.17 mmol L -1 (5050 mg L -1 ), while Z in 1 experienced time variations, as reported in Table 3. Moreover, Z in 2 , the corresponding concentrated NaOH solution, had a concentration of 18,870 mEq L -1 . Notice that since the main control objective was alkalinity and, eventually, the VFA concentration, only the influent composition related to these variables was included in Table 3. Moreover, as the VFA chemical equilibrium constants (and thus pKs) are very close, the AM2 dynamic model only takes into account the more representative of them as the model VFA molecule; i.e., acetic acid, whose chemical and physicochemical properties may represent all of the VFA ones. No other compounds, (i.e., weak acids or bases other than VFA and bicarbonates) affecting alkalinity were found in the red wine vinasses used in this work.

Results and Discussion

The design of the MIMO controller is based on adaptive robust exponential approaches [START_REF] Alcaraz-González | Robust Regulation of Alkalinity in Highly Uncertain Continuous Anaerobic Digestion Processes[END_REF] in order to provide synergy with the microbial growth and activity that have an exponential behavior. Thus, in the same way, the control law also provides synergy to S 2 production-consumption, and biogas production (mainly CO 2 ), with the VFA and bicarbonates being the main factors responsible of buffer system capacity, i.e., alkalinities.

As previously described, the control objectives, i.e., the set-points, were first established based on normal operational conditions stated by the alkalinity criteria [START_REF] Mata-Alvarez | A critical review on anaerobic co-digestion achievements between 2010 and 2013[END_REF][START_REF] Rehman | Anaerobic digestion[END_REF], and then formally expressed in the form of state variables based on a physicochemical analysis as in Equations ( 8) and ( 9). Since these are functions of pH, which varies with time, then the classical regulation control problem, with constant set-points, shifted into a tracking control problem, with variable set-points instead, which added robustness against pH variations. Figures 5 and6 show the behavior of S 2 and Z around their variable set-point calculated by using Equations ( 8) and ( 9) and pH measurements (Figure 9). Clearly, the MIMO control satisfactorily tracked the S 2 and Z set-point trajectories while satisfying the theoretical alkalinity criteria and preventing VFA accumulation. Thus, even when the alkalinity criteria [START_REF] Mata-Alvarez | A critical review on anaerobic co-digestion achievements between 2010 and 2013[END_REF][START_REF] Rehman | Anaerobic digestion[END_REF] are not involved explicitly or directly in the calculation of the MIMO control law given by Equations ( 10) and ( 11), this control law was able to satisfy indirectly the demanded constant reference inequalities of the alkalinities (3-4) in the system.

Processes 2021, 9, x FOR PEER REVIEW bicarbonates being the main factors responsible of buffer system cap i.e., alkalinities.

As previously described, the control objectives, i.e., the set-points first established based on normal operational conditions stated b alkalinity criteria [START_REF] Mata-Alvarez | A critical review on anaerobic co-digestion achievements between 2010 and 2013[END_REF][START_REF] Rehman | Anaerobic digestion[END_REF], and then formally expressed in the form o variables based on a physicochemical analysis as in Equations ( 8) an Since these are functions of pH, which varies with time, then the cla regulation control problem, with constant set-points, shifted into a tra control problem, with variable set-points instead, which added robu against pH variations. Figures 5 and6 show the behavior of 𝑆𝑆 2 a around their variable set-point calculated by using Equations ( 8) a and pH measurements (Figure 9). Clearly, the MIMO control satisfac tracked the 𝑆𝑆 2 and 𝑍𝑍 set-point trajectories while satisfying the theo alkalinity criteria and preventing VFA accumulation. Thus, even wh alkalinity criteria [START_REF] Mata-Alvarez | A critical review on anaerobic co-digestion achievements between 2010 and 2013[END_REF][START_REF] Rehman | Anaerobic digestion[END_REF] are not involved explicitly or directly i calculation of the MIMO control law given by Equations ( 10) and (11 control law was able to satisfy indirectly the demanded constant refe inequalities of the alkalinities (3-4) in the system. This can be seen in Figures 7 and8, which complete the picture implementation of the MIMO controller on the alkalinity variables, 𝐼𝐼 and 𝑇𝑇𝑇𝑇 , whose responses, although stable, did not mee predetermined bounds. The behavior of both these variables under c loop operation may be improved by changing the bounds o operational stability criteria. Another possible explanation will be later in this section.

On the other hand, uncertainties given for the lack of knowled This can be seen in Figures 7 and8, which complete the picture of the implementation of the MIMO controller on the alkalinity variables, I A/TA and TA, whose responses, although stable, did not meet the predetermined bounds. The behavior of both these variables under closed-loop operation may be improved by changing the bounds on the operational stability criteria. Another possible explanation will be given later in this section.

On the other hand, uncertainties given for the lack of knowledge on microbial growth kinetic is aggravated in AD systems by the impossibility of measuring biomass concentrations. However, these uncertainties were well addressed with the use of the Luenberger observer [START_REF] Godon | Microbial 16S rDNA diversity in an anaerobic digester[END_REF], which estimated η and S 2 . Figure 5 shows only the measurement of S 2 . and its reference given by Equation ( 8). This estimate is not shown here as a 99% precision was practically obtained. Besides, what it was actually used in Equation ( 3) was the direct measurements of S 2 instead of their estimates.

Figure 10 depicts the estimation of η, which was held below zero for most of the experimental run except in very brief time periods at t = 0.5 d, when both the Luenberger observer and the control algorithm were applied, and at t = 5.9 d and t = 8.9 d, when technical problems of the sensors and actuators occurred. It is important to remark that estimates of η < 0 means that the S 2 consumption rate is faster than its production, and so, substrate inhibition is unlikely to occur.

Processes 2021, 9, x FOR PEER REVIEW 13 of 19 reference given by Equation ( 8). This estimate is not shown here as a 99% precision was practically obtained. Besides, what it was actually used in Equation ( 3) was the direct measurements of 𝑆𝑆 2 instead of their estimates. Figure 10 depicts the estimation of 𝜂𝜂, which was held below zero for most of the experimental run except in very brief time periods at t = 0.5 d, when both the Luenberger observer and the control algorithm were applied, and at t = 5.9 d and t = 8.9 d, when technical problems of the sensors and actuators occurred. It is important to remark that estimates of 𝜂𝜂 < 0 means that the 𝑆𝑆 2 consumption rate is faster than its production, and so, substrate inhibition is unlikely to occur. Actuator control efforts for both control inputs, i.e., process input flow according to Equation [START_REF] Mir | Design considerations and operational performance of anaerobic digester: A review[END_REF] and additional input of concentrated alkali flow given by the supplementary NaOH supply according to Equation [START_REF] Holm-Nielsen | The future of anaerobic digestion and biogas utilization[END_REF], are shown in Figures 11 and12, respectively. Moreover, it can be seen that the bioreactor was stopped for sensor maintenance several times for less than one day at these short time periods (see Figures 1,[START_REF] Metcalf | Wastewater Engineering: Treatment, Disposal, and Reuse[END_REF][START_REF] Steyer | Evaluation of a four year experience with a fully instrumented anaerobic digestion process[END_REF][START_REF] Zhang | Characterization of food waste as feedstock for anaerobic digestion[END_REF]. Even in this extra-uncertain scenario, once the system was reestablished, the Luenberger observer as well as the proposed MIMO robust adaptive control approach recovered almost immediately their high performance, as seen in Figures 10,[START_REF] Holm-Nielsen | The future of anaerobic digestion and biogas utilization[END_REF] Actuator control efforts for both control inputs, i.e., process input flow according to Equation [START_REF] Mir | Design considerations and operational performance of anaerobic digester: A review[END_REF] and additional input of concentrated alkali flow given by the supplementary NaOH supply according to Equation [START_REF] Holm-Nielsen | The future of anaerobic digestion and biogas utilization[END_REF], are shown in Figures 11 and12, respectively. Moreover, it can be seen that the bioreactor was stopped for sensor maintenance several times for less than one day at these short time periods (see Figures 1,[START_REF] Metcalf | Wastewater Engineering: Treatment, Disposal, and Reuse[END_REF]5 and 6). Even in this extra-uncertain scenario, once the system was reestablished, the Luenberger observer as well as the proposed MIMO robust adaptive control approach recovered almost immediately their high performance, as seen in Figures 101112, demonstrating its robustness against sensors faults. However, there were differences between the values calculated by the MIMO control law and the actual flows given by the respective pumps. These differences appear to be more significant in the case of process input flow and less significant in the case of NaOH supply. Nevertheless, since the alkali solution was concentrated, the difference in this flow rate also has special relevance. Such differences may explain why 𝑇𝑇𝑇𝑇 𝑟𝑟 and (𝐼𝐼𝑇𝑇 𝑇𝑇𝑇𝑇 ⁄ ) 𝑟𝑟 were not strictly satisfied. Indeed, with such a high actuator sensitivity, it was not possible to achieve the flows demanded with the due accuracy. This fact inferred and led the system under conditions slightly different from what was expected, as shown in Figures 5678. Another possible factor, which in fact is in turn derived from the former, is that because of these technical difficulties, and because it was an online experimental validation (with all that this entails), the regulation on 𝑍𝑍 and 𝑆𝑆 2 could be achieved only in a neighborhood around their respective set-points. This regulation was relatively very close to the desired one, especially in 𝑆𝑆 2 . If it is taken into account that 𝐼𝐼𝑇𝑇 is mainly due to VFA by However, there were differences between the values calculated by the MIMO control law and the actual flows given by the respective pumps. These differences appear to be more significant in the case of process input flow and less significant in the case of NaOH supply. Nevertheless, since the alkali solution was concentrated, the difference in this flow rate also has special relevance. Such differences may explain why TA r and (I A/TA) r were not strictly satisfied. Indeed, with such a high actuator sensitivity, it was not possible to achieve the flows demanded with the due accuracy. This fact inferred and led the system under conditions slightly different from what was expected, as shown in Figures 5678. Another possible factor, which in fact is in turn derived from the former, is that because of these technical difficulties, and because it was an online experimental validation (with all that this entails), the regulation on Z and S 2 could be achieved only in a neighborhood around their respective set-points. This regulation was relatively very close to the desired one, especially in S 2 . If it is taken into account that I A is mainly due to VFA by 80% (at pH = 7) [START_REF] Ripley | Improved alkalimetric for anaerobic digestion wastes monitoring of municipal sludge[END_REF], this would help to explain how the regulation on I A/TA was relatively more performing than on TA. Furthermore, TA r was established above 60 mEq L -1 , and TA was effectively held in the whole experiment above this value. This feature together with the relatively good I A/TA regulation may explain the overall good performance of the system during this experimental run. Besides, in Figures 13 and14, the system behavior of the other variables, such as COD and gaseous flow rate Q gas , can be seen. Even though these variables were not expressly regulated, they remained in values both appropriate and suitable as COD < 3 g L -1 , and Q gas around 50 L h -1 on average.

Processes 2021, 9, x FOR PEER REVIEW 15 of 19 80% (at pH = 7) [START_REF] Ripley | Improved alkalimetric for anaerobic digestion wastes monitoring of municipal sludge[END_REF], this would help to explain how the regulation on 𝐼𝐼𝑇𝑇 𝑇𝑇𝑇𝑇 ⁄ was relatively more performing than on 𝑇𝑇𝑇𝑇. Furthermore, 𝑇𝑇𝑇𝑇 𝑟𝑟 was established above 60 mEq L -1 , and 𝑇𝑇𝑇𝑇 was effectively held in the whole experiment above this value. This feature together with the relatively good 𝐼𝐼𝑇𝑇 𝑇𝑇𝑇𝑇 ⁄ regulation may explain the overall good performance of the system during this experimental run. Besides, in Figures 13 and14, the system behavior of the other variables, such as COD and gaseous flow rate 𝑄𝑄 𝑔𝑔𝑚𝑚𝑔𝑔 , can be seen. Even though these variables were not expressly regulated, they remained in values both appropriate and suitable as COD < 3 g L -1 , and 𝑄𝑄 𝑔𝑔𝑚𝑚𝑔𝑔 around 50 L h -1 on average.

.Figure 13. Chemical oxygen demand in the digester. Furthermore, notice that the bioreactor reaches four stable regions induced by the four references stated in Table 1. It is pointed out that the proposed MIMO controller was implemented to drive the operating conditions to satisfy certain bounds on 𝑇𝑇𝑇𝑇 and the ratio 𝐼𝐼𝑇𝑇 𝑇𝑇𝑇𝑇 ⁄ , and the involved state variables 𝑆𝑆 2 and 𝑍𝑍. The behavior of other variables, such Furthermore, notice that the bioreactor reaches four stable regions induced by the four references stated in Table 1. It is pointed out that the proposed MIMO controller was implemented to drive the operating conditions to satisfy certain bounds on TA and the ratio I A/TA, and the involved state variables S 2 and Z. The behavior of other variables, such as COD and Q gas , remained within reasonable operation bounds. The actual manipulated variables Q in1 and Q in2 (proportional to D 1 and D 2 , respectively) showed no saturation during the whole experimental run.

Another aspect that can be discussed, is the hydraulic retention time (HRT) observed along the experiment and its relationship with OS issues. Notice that, with a 15 L h -1 average flow rate (see Figure 11) and a volume V = 0.982 m 3 , the average HRT = 2.72 d. Moreover, notice that, in this work, we have carried out four set-point changes (see Table 1), and each one had a time duration T ≤ HRT, except in the time interval 3.8 < t < 6, but in all cases just in time for achieving a steady state on the controlled variables, and to switch to the following one. In this way, the proposed MIMO control approach proved robustness and usefulness in real experimental implementation on an anaerobic digestion process used for the treatment of agro-industrial wastewater.

Finally, it should be noticed that a wholly operationally stable, steady-state operation could be widely desirable. Such an objective is possible, provided that the MIMO control objectives are extended properly. On one hand, reference values (8) and (9), i.e., S r 2 and Z r , are functions of the constant alkalinity references, but in turn, they are also functions of pH, which can vary in time as a function of environmental factors, such as temperature, and other ones. As a consequence, S r 2 and Z r are varying. Notice, however, that the true control objectives are TA and the ratio I A/TA. In this sense, by fulfilling these control objectives, OS is achieved, which was the global objective of the present work. Nevertheless, if a more ambitious objective is intended, i.e., to keep the system under OS, and at the same time to regulate other variables such as COD, gas production, or the same VFA concentration, the MIMO strategy should be adapted as a consequence. For instance, using the same bioreactor and similar control approaches, VFA regulation only [START_REF] Méndez-Acosta | Robust Control of Volatile Fatty Acids in Anaerobic Digestion Processes[END_REF] or COD regulation only [START_REF] Alcaraz-González | Robust interval-based regulation for anaerobic digestion processes[END_REF] already have been studied. In addition, in this work, the simultaneous regulation of alkalinities and VFA was addressed. Thus, the simultaneous regulation of alkalinities, VFA, and COD should be the next step in order to achieve both effective organic decontamination and efficient OS conditions.

Conclusions and Future Work

A multivariable control approach was proposed to regulate the alkalinities in anaerobic digestion processes and successfully implemented on an anaerobic 0.982 m 3 up-flow fixedbed bioreactor for the treatment of agro-industrial wastewater located in Narbonne, France, over 12 days in which several alkalinity set-points were tested. It was shown that such a control scheme satisfactorily tracks the set-point trajectories of the alkalinities in the face of time-varying pH, parameter uncertainties, and unknown microbial kinetics, while preventing the AD process to breakdown due to excessive accumulation of VFAs. The results also showed that this control scheme may be easily scaled-up and applied in actual wastewater treatment plants. Simultaneous automatic control of alkalinities, VFA, and COD in a wholly MIMO control approach is currently investigated as future work.

Figure 1 .

 1 Figure 1. Different values for 𝑍𝑍 𝑟𝑟 with (𝐼𝐼𝑇𝑇 𝑇𝑇𝑇𝑇 ⁄ ) 𝑟𝑟 = 0.3.

Figure 2 .

 2 Figure 2. Different values for 𝑆𝑆 2 𝑟𝑟 with 𝑇𝑇𝑇𝑇 𝑟𝑟 = 60 mmol L -1.

Figure 1 .

 1 Figure 1. Different values for Z r with (I A/TA) r = 0.3.

Figure 1 .

 1 Figure 1. Different values for 𝑍𝑍 𝑟𝑟 with (𝐼𝐼𝑇𝑇 𝑇𝑇𝑇𝑇 ⁄ ) 𝑟𝑟 = 0.3.

Figure 2 .

 2 Figure 2. Different values for 𝑆𝑆 2 𝑟𝑟 with 𝑇𝑇𝑇𝑇 𝑟𝑟 = 60 mmol L -1.Figure 2. Different values for S r 2 with TA r = 60 mmol L -1.

Figure 2 .

 2 Figure 2. Different values for 𝑆𝑆 2 𝑟𝑟 with 𝑇𝑇𝑇𝑇 𝑟𝑟 = 60 mmol L -1.Figure 2. Different values for S r 2 with TA r = 60 mmol L -1.

1 2

 1 (y(t)-y r (t))2

19 Figure 3 .

 193 Figure 3. Schematic representation of the proposed MIMO control approach.

Figure 3 .

 3 Figure 3. Schematic representation of the proposed MIMO control approach.

Figure 4 .

 4 Figure 4. Schematic representation of the anaerobic digestion process and its instrumentation (adapted from [5]).

Figure 5 .

 5 Figure 5. Volatile fatty acids concentration: (--) 𝑆𝑆 2 𝑟𝑟 , (─) 𝑆𝑆 2 (actual).

Figure 4 .

 4 Figure 4. Schematic representation of the anaerobic digestion process and its instrumentation (adapted from [5]).

Figure 4 .

 4 Figure 4. Schematic representation of the anaerobic digestion process and its instrumentation (adapted from [5]).

Figure 5 .

 5 Figure 5. Volatile fatty acids concentration: (--) 𝑆𝑆 2 𝑟𝑟 , (─) 𝑆𝑆 2 (actual).

Figure 5 .

 5 Figure 5. Volatile fatty acids concentration: (-) S r 2 , (-) S 2 (actual).

Figure 5 .

 5 Figure 5. Volatile fatty acids concentration: (--) 𝑆𝑆 2 𝑟𝑟 , (─) 𝑆𝑆 2 (actual).

Figure 6 .

 6 Figure 6. Strong ions concentration: (--) 𝑍𝑍 𝑟𝑟 , (─) 𝑍𝑍 (actual).

Figure 6 .Figure 7 .

 67 Figure 6. Strong ions concentration: (-) Z r , (-) Z (actual).

Figure 7 .

 7 Figure 7. Alkalinities' relationship and its regulation: (-) (I A/TA) r , (-) I A/TA (actual).

Figure 7 .

 7 Figure 7. Alkalinities' relationship and its regulation: (--) (𝐼𝐼𝑇𝑇 𝑇𝑇𝑇𝑇 ⁄ ) 𝑟𝑟 , (─) 𝐼𝐼𝑇𝑇 𝑇𝑇𝑇𝑇 ⁄ (actual).

Figure 8 .

 8 Figure 8. Total alkalinity regulation: (--) 𝑇𝑇𝑇𝑇 𝑟𝑟 , (─) 𝑇𝑇𝑇𝑇 (actual).

Figure 8 .

 8 Figure 8. Total alkalinity regulation: (-) TA r , (-) TA (actual).

Figure 9 .

 9 Figure 9. pH in the digester.

Figure 9 .

 9 Figure 9. pH in the digester.

Figure 10 .

 10 Figure 10. Estimate of 𝜂𝜂 given by the Luenberger observer.

  , and 12, demonstrating its robustness against sensors faults.

Figure 10 .

 10 Figure 10. Estimate of η given by the Luenberger observer.

Figure 11 .

 11 Figure 11. Input flow rate: (─) 𝛼𝛼 � 1 (calculated), (─) 𝛼𝛼 1 (actual).Figure 11. Input flow rate: (-) D1 (calculated), (-) D 1 (actual).

Figure 11 .

 11 Figure 11. Input flow rate: (─) 𝛼𝛼 � 1 (calculated), (─) 𝛼𝛼 1 (actual).Figure 11. Input flow rate: (-) D1 (calculated), (-) D 1 (actual).

Figure 11 .

 11 Figure 11. Input flow rate: (─) 𝛼𝛼 � 1 (calculated), (─) 𝛼𝛼 1 (actual).

Figure 12 .

 12 Figure 12. Alkali input flow rate: (─) 𝛼𝛼 � 2 (calculated), (─) 𝛼𝛼 2 (actual).

Figure 12 .

 12 Figure 12. Alkali input flow rate: (-) D2 (calculated), (-) D 2 (actual).

Figure 13 ..Figure 13 .

 1313 Figure 13. Chemical oxygen demand in the digester.

Figure 14 .

 14 Figure 14. Produced gaseous flow rate.

Figure 14 .

 14 Figure 14. Produced gaseous flow rate.

  • C and it could change ±1 • C around 35 • C.

Processes 2021, 9, x FOR PEER REVIEW 9 of 19

Table 1 .

 1 Values used as alkalinity references.

	Period T (d)	3.3	2.2	2.7	3.3
	TA r (mEq L -1 ) (I A/TA) r	90 0.30	90 0.28	85 0.27	75 0.30

Interval Time (d) 0.5 < t < 3.8 3.8 < t < 6 6 < t < 8.7 8.7 < t < 12

Table 2 .

 2 Values used in the Luenberger observer[START_REF] Godon | Microbial 16S rDNA diversity in an anaerobic digester[END_REF].

	Γ	g 1	g 2
	0.7	2	1

Table 3 .

 3 Values used as the process input concentrations.

	Interval Time (d)	0.5 < t < 6	6 < t < 8.7	8.7 < t < 12
	S in 2 (mmol L -1 ) Z in 1 (mEq L -1 ) Z in 2 (mEq L -1 )	84.17 90 18,870	84.17 80 18,870	84.17 75 18,870

Processes 2021, 9, 1153. https://doi.org/10.3390/pr9071153 https://www.mdpi.com/journal/processes

Processes 2021,[START_REF] Angelidaki | Applications of the Anaerobic Digestion Process[END_REF] 1153

Processes 2021, 9, x FOR PEER REVIEW

Funding: This research received no external funding.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement:

The data presented in this study are available on request from the corresponding author.

Conflicts of Interest:

The authors declare no conflict of interest.


Tom Druet

Arnaud Recoquillay

Simon Nehr

Margaux Horpin

Olivier Mesnil

Bastien Chapuis

Guillaume Laffont

Oscar D'almeida

GUIDED WAVE IMAGING OF A COMPOSITE PLATE USING PASSIVE ACQUISITONS BY FIBER BRAGG GRATINGS ON OPTICAL FIBERS

Keywords: Passive imaging, Fiber Bragg Gratings, elastic guided waves, Structural Health Monitoring

In this work are presented imaging results of defects in composite panels using guided wave imaging algorithms with Fiber Bragg Gratings (FBG) receivers and no piezoelectric actuation. Guided wave imaging is commonly applied to active data: each sensor acts successively as a source whereas other sensors measure the propagated signals. Here the imaging is obtained from passive data, that is, the signal corresponding to each emitter-receiver couple is reconstructed thanks to the crosscorrelation of the ambient noise measured simultaneously by every couple of sensors. The passive imaging approach allows the use of sensors unable to generate guided waves, namely FBG in optical fibers. FBGs present a much smaller intrusiveness compared to traditional piezoelectric sensors while allowing dense multiplexing and are much more robust to harsh environments. Results are shown for two instrumentations, the first one using piezoelectric transducers for actuation and FBG for reception while the second one uses FBG sensors only. The imaging results presented here show the feasibility of passive imaging in composite panels and the possibility of using FBG at ultrasonic frequencies, reducing the intrusiveness of the sensors integrated in the structure in the context of Structural Health Monitoring (SHM).

INTRODUCTION

Structural Health Monitoring (SHM) relies on the permanent integration of sensors. As such, the sensors must be particularly efficient, resilient and lightweight. In Guided Wave (GW) SHM, piezoelectric sensors are very often used because of their affordability and simplicity of use. Piezoelectric sensors are fairly light due to their size but as each must be cabled individually, the wiring quickly leads to non-negligible masses. Moreover, piezoelectric sensors work with a limited range of temperatures and even though high-temperature piezoelectric material do exist, significant maturation of the technology remains to be done. One of the most promising approach to overcome these limitations is to replace the piezoelectric sensors by Fiber Bragg Gratings (FBGs) on optical fibers. FBGs are short periodic inscription made inside the optical fiber core reflecting a specific wavelength. While stretched or compressed, the reflected wavelength changes, thus creating a strain sensor. Because of this highly dynamic phenomenon and a high sensitivity, FBGs can be used to measure elastic waves with sampling frequencies of the order of the mega-hertz [START_REF] Betz | Acousto-ultrasonic sensing using fiber bragg gratings[END_REF][START_REF] Druet | Passive guided waves measurements using fiber Bragg gratings sensors[END_REF]. The main downside of the FBGs compared to the piezoelectric sensors is their inability to generate a strain; the ultrasonic energy must therefore come from elsewhere for GW-SHM inspection. Hybrid piezoelectric -FBG systems have been proposed to limit the number of piezoelectric sensors (therefore the mass of the instrumentation), but fully passive systems using only FBGs are also possible. Passive systems in SHM relies on the interpretation of the vibrational noise naturally present in a structure to conduct the inspection. Under some assumptions, GW wavepackets equivalent to active signals (that is, the signal that would be generated by a transducer) can be reconstructed. In this paper, a fully passive GW-SHM system is presented, first hybrid with piezoelectric sensors and FBGs and second with FBGs only. To generate uncontrolled vibrational noise in the inspected systems representative of an inflight inspection, compressed air is sprayed on the surface of the studied composite panel. Imaging is conducted with the Excitelet algorithm [START_REF] Quaegebeur | Dispersion-based imaging for structural health monitoring using sparse and compact arrays[END_REF][START_REF] Kulakovskyi | Defect imaging on CFRP and honeycomb composite structures by guided waves generated and detected by a sparse PZT array[END_REF]

Passive acquisition by cross-corelation of ambient noise

In this section, the method to reconstruct active-like signals from passive measurements is summarized. As a reminder, the passive acquisition is defined by the signals acquired by the sensors while the system is exposed to an uncontrolled solicitation, in our case air flow. Because it contains very little relevant information, a passive acquisition is much longer than an active one, of the order of the second. In the following, the ambient noise φ is a space-time stationary random field that is also delta correlated in space and time. Consider a plate as a 2D domain and consider two sensors placed at positions x and y. Let us denote 𝑉 x , respectivelyV y , the passive signals acquired by the sensor at x, respectively y, on the time interval [0, 𝑇], 𝑇 > 0. The cross-correlation C T of these two signals is given by:

where H xy is the impulse response of the system formed by the plate and the sensors, the sensor at x acting as a source while the one at y acts as a receiver. Note that if the domain is reciprocal, which is the case for composite plates, then H xy = H yx . As the impulse response is causal, the positive time part of (3) corresponds to the impulse response for 𝑥 as a source and 𝑦 as a receiver and the negative part to the time-reversed impulse response for 𝑦 as a source and 𝑥 as a receiver. In other words, the time derivative of the correlation between the passive signals measured at both sensors is proportional to the impulse response from one sensor to the other. By doing this operation, virtual sources are created at the position of each sensor, and GW Imaging (GWI) algorithms can be used. This process is conducted for the whole frequency spectrum of the uncontrolled excitation.

Imaging results

To create signals for the imaging process, the passive signals are deconvolved over a frequency band of interest. In the configuration presented herein, the deconvolution is conducted to reconstruct signals equivalent to five-cycle and two-cycle tone burst centered at 30 kHz. The experiment was carried out on a 1010x610x2.2 mm 3 carbon fiber reinforced polymer flat panel with 8 plies all oriented in the same direction. It is instrumented on both sides, one side with piezoelectric sensor and one side with FBGs, both distributed on the same 30 cm diameter circle, so that FBGs and piezoelectric sensors are superimposed on either side. The wave sensing by the FBGs being directionally dependent, these sensors are here disposed radially to ensure maximal sensitivity to the area within the circle of sensors. The defect is simulated with a pair of circular magnets of radius 16 mm to emulate a guided wave scatterer. The GWI method is Excitelet [START_REF] Quaegebeur | Dispersion-based imaging for structural health monitoring using sparse and compact arrays[END_REF][START_REF] Kulakovskyi | Defect imaging on CFRP and honeycomb composite structures by guided waves generated and detected by a sparse PZT array[END_REF] and relies on the residual signals, defined as the difference between the pristine signals and the signals acquired in the unknown state. The residual signals between every pair of sensors are then compared for every combination of sensors to the theoretical ones for every possible position of the defect. Excitelet assumes the defect to be a point-like omnidirectional defect and requires the knowledge of the dispersion curves.

Because the dispersion is taken into account, a wider frequency spectrum brings better resolved results than a narrow frequency spectrum.

The imaging is conducted for both the hybrid configuration with piezoelectric sensors as emitters and FBGs as receivers as reported in Figure 1 and in the passive configuration with the FBG sensors only as reported in Figures 2 and3. In the active hybrid configuration (Figure 1), the piezoelectric sensors are excited by a 5-cycle tone burst. In the passive configuration of In the passive configuration, since the signals must be acquired at the same time to conduct the correlation, the number of channels of the acquisition chain is a limit. In our case, the FBG acquisition system was limited to 4 channels in parallel. In practice, the passive acquisition was conducted twice for 4 FBGs sensors. This means that only 24 couple of FBG sensors are available, against 56 for the active acquisition, therefore the quality of the two images is not directly comparable. This is why even though the defect is properly detected and located in both Figures 1 and2, the spot for the defect in the image obtained by the FBGs is bigger. Figure 2 however demonstrates the feasibility to obtain excellent GWI results for this configuration with FBGs and an uncontrolled vibrational solicitation. Figure 3 shows also detection and localization with a thinner spot on the true defect position. This is due to the fact that a broader frequency spectrum is used for the Excitelet imaging therefore better resolution is obtained. The vertical and horizontal lines in Figure 3 are mainly due to the fact that 2 pairs of 4 FBGs are measured in parallel, creating these imaging artifacts. The artifacts are expected to disappear if 8 FBGs could be measured in parallel leading to 56 pairs of sensors instead of 24.

Conclusion

This paper demonstrates the feasibility to conduct GW-SHM inspections with FBG sensors only and without any controlled GW excitation. It is demonstrated that similar results are obtained with a controlled GW excitation by piezoelectric transducers and uncontrolled GW excitation with compressed air sprayed on the surface.

The ambition is to be able to conduct in-service inspections using the natural vibration of the structure during its exploitation. For aerospace for example, the aerodynamic flow or the rotating motors would provide the necessary vibrations for the inspection. The use of FBGs is a significant breakthrough in SHM as they reduce significantly the added mass and intrusiveness of the overall system, and expand the field of application of such technologies to higher temperatures or radioactive environments.

2 ©

 2 2019 by ASME 𝐶 𝑇 (𝑡, 𝑥, 𝑦) = 1 𝑇 ∫ 𝑉 𝑥 (𝜏)𝑉 𝑦 (𝑡 + 𝜏) d𝜏 𝑇 0 . (1) One can then show [5] that d 𝐶 𝑇 d𝑡 ∝ 𝐻 𝑥𝑦 (𝑡) -𝐻 𝑥𝑦 (-𝑡),

Figures 2 and 3 ,

 3 Figures 2 and 3, the passive signals are deconvolved over a 5 and 2-cycle 30 kHz tone burst respectively.

FIGURE 1 :

 1 FIGURE 1: Active hybrid imaging (Emission by piezoelectric, reception by FBGs) of a defect represented by the white circle in a CFRP plate using the Excitelet algorithm and 8 piezoelectric sensors. The excitation is a 5-cycle tone burst at 30 kHz.

FIGURE 2 :FIGURE 3 :

 23 FIGURE 2: Passive imaging (vibration generated by compressed air, reception by FBGs) of a defect represented by the white circle in a CFRP plate using the Excitelet algorithm and 8 FBG sensors. Passive signals are deconvolved to reconstruct a 5-cycle tone burst at 30 kHz


P André

M Benmouffok

M Chnani

P Freton

J J Gonzalez

P Joyeux

A Risacher

Fraction molaire pour les mélanges Air-AgC et Air-AgSnO 2 dans un plasma à et hors de l'équilibre thermique

Introduction

Méthode de calcul

Afin de calculer les fractions molaires, nous devons prendre en compte les propriétés thermodynamiques spécifiques de chaque espèce chimique. Afin de compléter les tables de thermochimie existantes (JANAF), nous avons calculé les propriétés spécifiques (enthalpie h, entropie s, potentiel chimique 𝜇 0 et capacité calorifique 𝑐 𝑝 ) de Sn 2 , SnO et SnO 2 .

j max =256 j max =129 j max =69 j max =176 j max =218 j max =324 j max =291 j max =355 j max =385 j max =413

  Dans le cadre d'un travail collaboratif soutenu par l'Association Arc Électrique (AAE), une « Étude expérimentale de la dynamique d'un arc électrique à courant continu évoluant entre deux rails » est menée. L'expérimentation est réalisée au laboratoire LAPLACE et plusieurs situations notamment celle d'un arc électrique généré entre deux contacts électriques sont envisagées. Lors de l'ouverture des contacts ou après l'ignition de l'arc électrique, la température augmente et les matériaux de contact changent d'état. S'ils se vaporisent, ils alimentent le plasma et se redéposent. Le plasma formé dans l'environnement immédiat des contacts électriques peut se trouver dans des états d'équilibre thermodynamique et de pression variés. Par conséquent, nous étudions les fractions molaires dans le cas de contacts constitués de pseudo-alliage, argent-graphite et argent-dioxyde d'étain, environnés d'air.

•

  La vaporisation du dioxyde d'étain a lieu à plus basse température que celle de l'argent. • Le graphite brûle si le milieu est en excès d'oxygène et forme du dioxyde de carbone.• Le dioxyde d'étain liquide se vaporise en monoxyde d'étain et en dioxygène.

• Plus la pression est grande plus les phases condensées sont présentes à haute température. • La température de vaporisation est peu sensible au déséquilibre thermique (θ trop faible). • On note la présence du dioxyde d'étain sous forme solide tandis que l'argent est sous forme liquide.

  𝑁 𝑔𝑎𝑧 et 𝑁 𝑐𝑜𝑛𝑑𝑒𝑛𝑠é𝑠 représentent respectivement le nombre d′espèce chimique en phase gazeuse et en phase condensée.

	𝐺 = 0 (gaz) représente le potentiel chimique de l′espèce j en phase gazeuse 𝑗=1 𝑁 gaz 𝑛 𝑗 𝜇 𝑗 0 (gaz) + i=1 𝑁 condensés 𝑛 i 𝜇 i 0 (liquide_solide) avec 𝜇 𝑗 𝜇 𝑖 0 (liquide_solideRésultats 0.0001 0.001 0.01 0.1 1 2000 4000 Ag NO SnO Ag 2 CO 2 Ar SnO 2 liquide SnO 2 cristallin Ag solide Ag liquide O 2 O N 2 Température (K) 6000 N Sn CO Fraction Molaire (Air, Ag, SnO 2 ) (50%, 45%, 5%) ; P=1 Bar 8000 Ag + N Sn Air Ag SnO 2 Air Ag C 10000 + O + + e -0.0001 0.001 0.01 0.1 1 CO 2 Ar SnO 2 cristallin 2000 SnO 2 liquide Ag solide Ag liquide Fraction Molaire (Air, Ag, SnO 2 ) (50%, 45%, 5%) ; P=10 Bars 4000 6000 8000 Ag -N Ag Ag + Sn + e -Sn NO SnO Ag 2 CO O 2 O N 2 Température (K) 0.0001 0.001 0.01 0.1 1 2000 4000 6000 8000 10000 Ag ++ Sn ++ NO + CO Ar + N Ag Ag + Sn + O + N + e -Sn NO SnO Ag 2 CO 2 Ar SnO 2 liquide SnO 2 cristallin Ag solide Ag liquide O 2 O N 2 Température des espèces lourdes (K) Fraction Molaire (Air, Ag, SnO 2 ) (50%, 45%, 5%) ; P=1 Bar ; =2	10000 N +
	Conclusion

) représente le potentiel chimique de l′ espèce i en phase cristalline ou liquide.

AgC: La non-miscibilité de l'argent et du graphite fait que les contacts de type AgC sont habituellement réalisés par frittage. Lors du passage du pied d'arc sur le contact électrique la combustion du graphite va avoir lieu laissant la surface du contact modifié, ce qui aura pour conséquence la modification des propriétés physique du contact. La proportion de graphite doit être suffisamment faible pour éviter son dépôt dans l'appareillage électrique. AgSnO 2 : La vaporisation du dioxyde d'étain ayant lieu avant la vaporisation de l'argent, l'argent se trouvant sous forme liquide, il aura tendance à diffuser plus facilement dans le pseudo-alliage jusqu'à la surface du contact modifiant les propriétés physiques notamment en augmentant la résistance du contact.

Au niveau du pied d'arc la pression peut atteindre des valeurs élevées, les températures de vaporisation augmentent. Ces premiers calculs présentent de nombreuses limites, notamment la structure du contact et des phénomènes internes non pris en compte: tension de surface liquide-solide (Ag, SnO 2 ), corrosion sèche, additifs, taille des poudres, diffusion de l'oxygène et oxydation sélective, incertitude sur les enthalpies de formation, cinétique temporelle … A partir de l'énergie libre de Gibbs G, la neutralité électrique et la conservation du nombre et du type de noyau atomique, nous pouvons déterminer les fractions molaires 𝑛 𝑗 en fonction de la température: