Software Sensor for Online Estimation of the VFA's Concentration in

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Process Systems Engineering

Software sensor for on-line estimation of the VFA's concentration in anaerobic digestion processes via a high-order sliding mode observer Gerardo Lara-Cisneros, and Denis Dochain Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b02607 • Publication Date (Web): 04 Oct 2018 Downloaded from http://pubs.acs.org on October 10, 2018

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Industrial & Engineering Chemistry Research

Software sensor for on-line estimation of the VFA's on entration in anaerobi digestion pro esses via a high-order sliding mode observer ∗

Gerardo Lara-Cisneros

and Denis Do hain

Mathemati al Engineering Department, ICTEAM, Université atholique de Louvain, 4-6 avenue G. Lemaître, 1348 Louvain-la-Neuve, Belgium. E-mail: glara isgmail. om

Abstra t This paper deals with the design of a software sensor based on a high-order sliding mode observer (HOSMO) for on-line estimation of the on entration of Volatile Fatty A ids (VFA's) in a lass of ontinuous anaerobi digestion pro esses (AD). Taking into a

ount the limited availability for on-line monitoring of AD pro esses, in this

ontribution it is assumed that only the methane outow rate is available for on-line measurement. Considering a simple two-dimensional AD model it is shown that the VFA on entration an be dete table from the on-line measurement of methane outow rate. The estimation method is based on a lo al oordinate transformation that

onverts the original model into an observable system. In the observable form a supertwisting algorithm is proposed in order to provide robustness under perturbation and un ertainty onditions. The observer onvergen e is analyzed by using Lyapunov stability te hniques. Numeri al simulations illustrate the ee tiveness of the proposed

1

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estimation s heme for a four-dimensional AD model with un ertainties asso iated with unmodeled dynami s and disturban es in the inow omposition.

Introdu tion Anaerobi Digestion (AD) has been gained attention in the last de ades, on one hand is a traditional organi waste treatment te hnology to redu e organi matter from agro-food industrial wastes and muni ipal euents and, on the other hand it plays an important role as a sour e of renewable energy, be ause AD produ es biomethane 1. From renewable energy sour e point of view, in reasing the methane produ tion is one of the key issues for the optimal operation of anaerobi digestion pro esses 24 . However the implementation of large-s ale AD plants has been limited due to severe operational problems to a hieve stable operation of the pro ess 59 . In fa t it is di ult to keep AD on the optimal operating

onditions, be ause of the potential instability indu ed by the inhibition in presen e of high levels of Volatile Fatty A ids (VFA) 6,7,10,11 . It is well known that the inhibition of the methanogeni ba teria growth rate by a

umulation of Volatile Fatty A ids (VFA) indu es a idi ation of the system and leads to the pro ess failure 6,7,11 . But at the same time VFA's are the limiting substrate for methane produ tion in the methanogenesis bio hemi al pro ess. 1 Consequently the on entration of VFA is a very important variable for operating and ontrolling AD pro esses 1 . Additionally, u tuation in the inlet omposition of the digester auses that the optimal operation of the AD pro ess to be very hard to keep on for open-loop onguration 12 . The design of ontrol s hemes for optimal operation of AD pro ess requires advan ed online measurement systems for an adequate monitoring the key pro ess variables 3,13 . However, the existing monitoring equipment for riti al variables of anaerobi pro esses su h as organi a id on entrations and the main ba terial populations are too expensive and require extensive maintenan e 14 . In pra ti e, it is di ult to measure all states required for a well 2


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monitoring and ontrol of AD pro ess, due to the la k or high pri e of sensors. Only few pro ess parameters su h as pH, temperature, redox potential, gas produ tion rate and ow rates are available in a ost ee tive manner for on-line measurement. 15 An interesting alternative is to use mathemati al models of the AD pro ess with a limited set of available measurements so that we have an estimation of the time evolution of the key pro ess variables by mean of state observers also known as software sensors (or soft sensors) 16,17 . The design of state observers in biopro esses has been an a tive area over the last de ades 1820 . Spe i ally for the AD pro ess, in literature we an nd dierent state estimation s hemes from lassi al Kalman lters and adaptive observers s hemes to nonlinear asymptoti and interval observers 13,14,18,2027 . The aim of most state observers proposed in literature is to provide the estimation for the key AD variables from the on-line measurement of the organi substrate on entration (expressed as hemi al oxygen demand) or the total organi fatty a ids on entrations (or both) 28 . However, in pra ti e the biogas ow rate an be more easily measured on-line that the organi substrates on entration or some spe i ba terial populations 26. In fa t, today the advan ed monitoring s hemes for internal states of the pro ess is only possible by mean of spe tros opy-based instrumentation equipments 15 . The on-line estimation of the key variables in AD pro esses when only the biogas outow rate is available for on-line measurement is an open issue in urrent literature. 29 Nevertheless, only a few works an be found in open literature with respe t to estimation s hemes of AD from biogas outow monitoring 21,23,25,26,29,30 . In Bernard et al. 21 an asymptoti observer has been proposed for estimation of COD and VFA from the on-line gaseous measurement. This lass of observers are based on a state transformation leading to a subsystem independent of the growth kineti s expressions. The main drawba k of the asymptoti observers is that it requires the perfe t knowledge of the yield oe ients (or a ratio of them), and may be very sensitive to unknown load disturban es 23 . More re ently, in Carlos-Hernández et al. 30 a ontrol strategy is proposed for bi arbonate regulation in AD pro ess based on a fuzzy ontroller with a Takagi-Sugeno observer omposed by 45 lo al observers. Also, in 25,26 3



Kalman-type observers have been proposed for the estimation of key variables in AD pro ess with only methane gas ow measurement. The main issue of the lo al observers approa hes is their poor performan e for operating ondition far from of the designed onditions, mainly for strongly nonlinear and intrinsi ally unstable systems 20 . In 29 an on-line estimator of the VFA on entration has been proposed from the measurement of methane outow rate. However, the main disadvantage of the above observer s heme is that it does not provide any guideline for the appropriate hoi e of the observer parameters and its performan e is very sensitive to those. 29 In this framework it is well known that dis ontinuous inje tion term of sliding mode observers improve the onvergen e properties in perturbation and un ertain

onditions 3133 . However in traditional sliding mode observers the dis ontinuous term may enhan e robustness but this an be at the expense of in reased sensitivity to measurement noise. In literature it is shown that higher order sliding mode te hniques an improve the performan e of the observer in high un ertain onditions 3436 . The high-order sliding mode is a generalization of traditional sliding modes in whi h an inje tion term a ts on higher derivatives of the sliding variable 35. Higher-order sliding mode observers (HOSMO) have been widely studied and developed for diverse appli ations in me hani al systems 35,36 . More re ently appli ations of the HOSMO for state and input estimation of biopro esses have been reported in 3739 . Nevertheless, as far as we know, there is not HOSMO s hemes for on-line estimation of AD pro ess with only the methane outow rate as the measurable signal. In this ontribution we propose a robust estimation methodology based on a high-order sliding mode observer for the on-line estimation of VFA on entration in a lass of ontinuous AD pro esses. The estimation obje tive in this proposal is the same as in the work, 29 i.e., the on-line estimation of the VFA's on entration in ontinuous AD pro esses from the measurement of the methane outow rate. However, the design methodology and the observer stru ture are very dierent as will be explained in the rest of the do ument. First, from a simple two-dimensional AD model it is shown via dierential algebrai approa h that the VFA on entration an be dete table from the on-line measurement of methane outow rate. 4


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The estimation s heme proposed is based on a lo al oordinate transformation allowing to transform the two-dimensional AD model in an observable system. Next, in the observable form a super-twisting algorithm is implemented to provide robustness under perturbation and un ertainty onditions. The observer onvergen e is analyzed by using Lyapunov stability te hniques. Numeri al simulations illustrate the ee tiveness of the proposed estimation method for a four-dimensional AD model with un ertainties asso iated with unmodeled dynami s and disturban es in the inow omposition. The rest of the paper is organized as follows: In the next se tion the AD mathemati al model used for the observer design is presented, also the problem statement and some issues related with the VFA dete tability are dis ussed. The se tion "Observer design" ontains the design methodology of the proposed observer s heme and its onvergen e properties are analyzed. Numeri al experiments that illustrate the performan e of the proposed estimation approa h are shown in Se tion "Numeri al veri ation". Some on luding remarks are dis ussed in last se tion.

Model des ription and problem statement Let us onsider a simple mathemati al model of the AD pro ess proposed in Andrews 40 that onsiders a single degradation stage of soluble organi substrate (S ) by methanogeni ba teria (X ). µ(·)x

kt S # X + km CH4

where kt is the yield oe ient asso iated with substrate degradation and µ(·) stands for methanogeni ba teria growth rate. The orresponding mass balan e for a ontinuous anaerobi pro ess reads: S˙ = u(Sf − S) − kt µ(·)X X˙ = µ(·)X − αuX

5


(1)


Page 6 of 30

where S and X represent the on entration of soluble organi substrate and methanogeni ba teria, respe tively; u is the dilution rate given by the ratio between the feeding ow with the volume of the digester; α is the fra tion of ba teria not atta hed onto a support (i.e., being ae ted by the dilution rate in the digester), and Sf is the on entration of the soluble organi substrate inlet in the digester. Be ause the solubility of methane in the liquid phase is very low, the on entration of dissolved methane is negle ted and the produ ed methane is assumed to go dire tly out of the digester, with the outow rate of methane gas Qm proportional to the growth rate of the methanogeni ba teria (see, 18 ) (2)

Qm = km µ(·)X

where km is the yield oe ient for the methane produ tion. With respe t to the spe i growth rate for the methanogeni populations µ(·), for the observer design purpose it is assumed to be des ribed by a monotoni and bounded fun tion of the substrate on entration, e.g. like the Monod kineti model, with the following property:

Property 1. For all s ∈ SD where SD = s ∈ R 0 ≤ s ≤ sm with sm < ∞, µ ∈ C ∞ and

µ < µ with µ < ∞ as the upper bound of µ.

The system (1-2) an be rewritten in the form x˙ = f (x, u)

(3)

y = h(x)

where x = [x1 , x2 ] ∈ R2+ with x1 = S and x2 = X representing the state ve tor. The input signal u ∈ R represents the dilution rate, and the output measurable signal is given by the methane outow rate Qm as. y = h(x) = km µ(x1 )x2

6


(4)

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The ve tor eld f : R2 × R → R2 takes the form 



 u(Sf − x1 ) − kt µ(x1 )x2  f (x, u) =   µ(x1 )x2 − αux2

(5)

On the observability and dete tability In order to analyze the observability properties of the system (3-5) we will give a brief ba kground to the algebrai dierential observability approa h 41,42 . A dierential eld K is a ommutative eld of hara teristi zero, whi h is equipped with a ˙ b˙ and dtd (ab) = ab+a ˙ b˙ . single derivation dtd : K → K su h that, for any a, b ∈ K , dtd (a+b) = a+

A onstant of K is an element c ∈ K su h that c˙ = 0. A dierentiable eld extension L/K is given by two dierential elds K , L, su h that the derivation of K ⊆ L is the restri tion to K of the derivation of L. An element of L is said to be dierentially algebrai over K if, and only if, it satises an algebrai dierential equation with oe ients in K . The extension L/K is said to be dierentially algebrai if, and only if, any element of L is dierentially algebrai over K .

Notation. We denote Khκi, where κ is a subset of the dierentiable subeld generated by K and κ. A ground eld is a eld k whi h is xed in a given situation, su h that everything takes pla e "over" k. Let k be a given dierential ground eld as a eld of fun tions. A system is a nitely generated dierential extension K/k. A dynami al system is a system where a nite subset u = (ui , ..., um) ⊂ K of ontrol variables has been distinguished, su h that the extension K/khui is dierentially algebrai . An input-output system is a dynami al system where a nite subset y = (y1 , ..., yp ) ⊂ K of output variables has been distinguished. This means that any element of K satises a dierential-algebrai equation with oe ients, that are rational fun tions over khu, yi in the omponents of u, y and a nite number of their time derivatives.

7



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Then an element χ in K is said to be algebrai ally observable with respe t to {u, y} if it is algebrai over the dierentiable eld khy, ui.

Denition 1.

42

A system variable χ ∈ K is said to be algebrai ally observable if, and only

if, it is algebrai over khy, ui, so that χ satises a dierential polynomial in terms of u, y and some of their time derivatives, i.e., P (χ, u, u, ˙ ..., u(k) , y, y, ˙ ..., y (l) ) = 0

with oe ients over khy, ui. The above denition is alled the Algebrai Observability Condition (AOC) and means that any state variable is algebrai over khy, ui, i.e., is an algebrai fun tion of the omponents of u, y and of a nite number of their derivatives. It is known that this denition is equivalent to the lassi observability rank ondition for systems of the form (3-5)(see 42 ). In order to analyze the AOC ondition for the system (3-5) we should nd a dierential polynomial in terms of x, u, y and their time derivatives. From (3)-(4) and (5) it is easy to see that a dierential equation of lowest order of x1 is the following x˙ 1 − Sf u + x1 u +

kt y=0 km

(6)

whi h shows that it is possible to estimate x1 while its dynami s remains stable in terms of u, y and their time derivatives. In previous work 43 it has been shown that the nominal system (3,5), with µ as a monotoni fun tion satisfying Property 1, admits a lo ally stable equilibrium for αu < µ¯, where µ¯ is the upper bound of µ. Then we an ensure that the system will be lo ally stable if αu < µ¯. Now the following assumption is formulated.

Assumption 1. The dilution rate u remains in the bounded interval 0 < u < α−1 µ¯. Hen e, if Assumption 1 holds then x1 = S will be dete table with respe t to the pair {u, y} dened in (3-5). 8


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Observer design From the theory of anoni al forms for nonlinear systems 42,44 it is known that every uniformly observable n-dimensional SISO system an be transformed via a lo al oordinates transformation into an observable normal form: z˙1 = z2 z˙2 = z3

.. . z˙n−1 = zn z˙n = ϕ(x, u) y = z1

For the ase of the AD model (3-5) we dene the following lo al oordinates transformation: 



 φ1 (x)  z = Φ(x) =   φ2 (x)

(7)

with φ1 = h(x)

(8)

φ2 = Lf h(x)

Where Lf h(x) represents the Lie derivative of the output s alar fun tion (4) with respe t to the ve tor eld (5). By al ulating the Lie derivative we have the new oordinates as: (9)

z1 = km µ(x1 )x2 z2 = km Sf µ′ (x1 ) − km ux1 x2 µ′ (x1 )kt km µ(x1 )µ′ (x1 )x22 + km µ2 (x1 )x2 − km αuµ(x1)x2

9



where µ(x1 ) =

µm x2 Ks +x2

Page 10 of 30

has the form of the Monod's kineti s model, µ′ (x1 ) denotes the deriva-

tive of µ with respe t to x1 given by µ′ (x1 ) =

µm Ks (Ks +x1 )2

and µ2 (x1 ) =

µ2m x21 (Ks +x1 )2

. In terms of

the new oordinates (9) we an write an equivalent system as: z˙1 = z2

(10)

z˙2 = ϕ(x) y = z1

where ϕ(x) = L2f h(x) = [km Sf ux2 µ′′ (x1 ) − km ux1 x2 µ′′ (x1 ) − km ux2 µ′(x1 )kt km x22 µ′ (x1 ) − kt km x22 µ′ (x1 ) + 2km x2 µ(x1 )µ′ (x1 ) − k2 αux2 µ′ (x1 )][u(Sf − x1 ) − kt µ(x1 )x2 ] + [km Sf uµ′ (x1 ) − km ux1 µ′ (x1 ) − 2kt km µ(x1 )x2 + km µ2 (x1 ) − km αuµ′ (x1 )][µ(x1 )x2 − αux2 ].

The inverse map is given by 

 x = Φ−1 (z) = 

φ−1 1 (z) φ−1 2 (z)

  

(11)

where

φ−1 1 (z)

=

φ−1 2 (z)

=

−Ks ((α + 1)uz1 + z2 )z1 − [Ks2 ((α + 1)uz1 + z2 )2 z12 − 4(kt /km Ks z13 − Ks Sf uz12 )((αu − µm )z12 + z1 z2 )]1/2 2[(αu − µm )z12 + z1 z2 ] [Ks + φ−1 1 (z)]z1

(12)

km µm φ−1 1 (z)

Now in the spirit of Moreno (2013) work, 33 we propose the following Super-Twisting Observer (STO).

Proposition 1. Let us onsider the system (3-5) that satises the Assumption 1, then the following dynami al system with the inverse map (11-12) is a robust observer able to estimate

10


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the original states x = [x1 , x2 ] ∈ R2 zˆ˙1 = zˆ2 + κ1 γϑ1 (e1 )

(13)

zˆ˙2 = κ2 γ 2 ϑ2 (e1 )

where zˆ is the estimation ve tor of z with e1 = z1 − zˆ1 . κ1,2 are hosen su h that the polynomial P (λ) = λ2 + κ1 λ + κ2 is Hurwitz, and γ > 0 is an observer gain that has to be sele ted large enough to guarantee the onvergen e of the observer. The inje tion terms are of the form: ϑ1 (e1 ) = λ1 |e1 |1/2 sign(e1 ) + λ2 |e1 |q sign(e1 ) ϑ2 (e1 ) =

(14)

λ21 sign(e1 ) + λ1 λ2 (q + 1/2)|e1 |q−1/2 sign(e1 ) + λ2 |e1 |2q−1 sign(e1 ) 2

where λ1 , λ2 > 0 are non negative onstants, not both zero, and q ≥ 12 is a real number. Proof : For purpose of the onvergen e analysis of the observer (13-14) let us onsider the following system: z˙1 = z2 + δ1 (t, z, u)

(15)

z˙2 = δ2 (t, z, u)

with y = h(x) = z1 and δ1 , δ2 representing un ertain terms related with unmodeled dynami s and perturbation on the system (3-5). We dene the estimation error ve tor as e = [e1 , e2 ]T with e1 = z1 − zˆ1 and e2 = z2 − zˆ2 , then the dynami s of the estimation error is given by e˙ 1 = e2 − κ1 γϑ1 (e1 ) + δ1 e˙ 2 = −κ2 γ 2 ϑ2 (e1 ) + δ2

11


(16)


Page 12 of 30

For onvenien e we dene the following state ve tor 



 ϑ1 (e1 )  ε=  e2

(17)

The time derivative of ε is given by 



h i  e2 − κ1 γϑ1 (e1 ) + δ1  ′ ε˙ = ϑ′1 (e1 )   = ϑ1 (e1 ) (A0 − Γκ0 C0 )ε + δ˜ −κ2 γ 2 ϑ1 (e1 ) + δ2

with ϑ′1 (e1 ) =

dϑ1 (e1 ) de1

(18)

, 











 γ 0   κ1   0 1  A0 =    , C0 = [1, 0] , Γ =   , κ0 =  0 γ2 κ2 0 0

and



 δ˜ = 

2|e1 |1/2 λ1 +2qλ2 |e1 |q−1/2



δ1   δ2

e=ϑ′1 (ε)

Note the following: rst, ϑ1 and ϑ2 are related, sin e ϑ2 (e1 ) = ϑ′1 (e1 )ϑ1 (e1 ), they are both monotoni ally in reasing fun tions of e1 and ϑ1 is ontinuous while ϑ2 is dis ontinuous at e1 = 0. Note that the hara teristi polynomial of the matrix (A0 − Γκ0 C0 ) is equal to p(s) = det|sI2 − (A0 − Γκ0 C0 )| = s2 + γκ1 s + γ 2 κ2 ; and, on the other hand, p(s) = s2 + γκ1 s + γ 2 κ2 = (s − γυ1 )(s − γυ2 ) where υ1 , υ2 are the eigenvalues of the (Hurwitz)

matrix Aκ = (A0 − κ0 C0 ), i.e. the matrix (A0 − Γκ0 C0 ) with γ = 1. This shows that the eigenvalues of (A0 − Γκ0 C0 ) are γυ1 , γυ2 , multiples of the eigenvalues of (A0 − κ0 C0 ). Similar to the lassi al proof method for High-gain observers 24 we introdu e here a hange of variables



 ξ = θΓ−1 ε = 

12

θ εγ1 θ γε22

  


(19)

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

where Γ−1 =  

1 γ

0



0   and θ > 0 is an arbitrary positive onstant, we obtain (sin e

1 γ2

Γ−1 A0 Γ = γA0 and C0 Γ = γC0 )

ξ˙ =

θΓ−1 ϑ′1 (e1 )

i h 1 ′ −1 ˜ ˜ (A0 − Γκ0 C0 ) Γξ + δ = ϑ1 (e1 ) γ(A0 − κ0 C0 )ξ + θΓ δ θ

(20)

Now, the following quadrati Lyapunov fun tion is proposed as V (ξ) = ξ T P ξ

(21)

where P = P T > 0 is a unique, symmetri and positive denite solution of the Algebrai Lyapunov Equation (ALE) (A0 − κ0 C0 )T P + P (A0 − κ0 C0 ) = −Q

(22)

for Q = QT > 0 an arbitrary positive denite and symmetri matrix. The derivative of V along the solutions of the error equation (20) is given by dV (ξ) = V˙ (ξ) = ξ T P ξ˙ + ξ˙T P ξ dt

repla ing (20) we have h i V˙ = ϑ′1 (e1 ) γξ T [(A0 − κ0 C0 )T P + P (A0 − κ0 C0 )]ξ + 2ξ T P θΓ−1 δ˜

Now by onsidering the ALE (22) h i ′ T T −1 ˜ ˙ V = ϑ1 (e1 ) −γξ Qξ + 2ξ P θΓ δ

From the standard inequality for quadrati forms λmin {P } kξk2 ≤ ξ T P ξ ≤ λmax {P } kξk2

13


(23)


Page 14 of 30

It follows that h i ˜ V˙ ≤ ϑ′1 (e1 ) −γλmin {Q} kξk2 + 2kξk kP k kθΓ−1δk

(24)

where λmin {Q} is the minimal eigenvalue of Q, kξk is the Eu lidean norm of ξ and kP k = λmax {P } is the indu ed (Eu lidean) norm of matrix P . Re all that ϑ′1 (e1 ) > 0 sin e ϑ1 (e1 )

is monotoni ally in reasing. Now, we will assume here that there are some onstants ρ1 , ρ2 su h that the perturbation term δ˜ satises the following restri tions

|δ˜1 | = |δ1 | ≤ ρ1 |ϑ1 (e1 )| = ρ1 (λ1 + λ2 |e1 |q−1/2 )|e1 |1/2 1/2 2|e | 1 2 2 |δ | ≤ ρ ϑ (e ) + e |δ˜1 | = 2 2 1 2 λ1 + 2qλ2 |e1 |q−1/2

Using the relations γθ ξ1 = ε1 = ϑ1 (e1 ) and

θ2 θ2 kθΓ δk = 2 δ˜12 + 4 δ˜22 ≤ γ γ −1 ˜ 2

(25)

= ε2 = e2 , we obtain the inequalities

γ2 ξ θ 2

θ2 2 2 1 2 2 2 2 θ2 ˜2 θ2 ˜2 2 2 δ1 + 4 δ2 ϑ1 (e1 )+ 4 ρ2 e2 = ρ1 + 2 ρ2 ξ1 +ρ2 ξ2 ≤ ρ2 kξk2 2 γ γ γ γ

for 2

ρ ≥ max

ρ21

1 + 2 ρ22 , ρ22 γ

This implies that V˙ ≤ −ϑ′1 (e1 ) [γλmin {Q} − 2ρλmax {P }] kξk2

So that V˙ ≤ 0 for a su ient large gain γ , e.g. γ > γ0 , 2ρ

14

λmax {P } λmin {Q}


Page 15 of 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


This an always be a hieved for a su ient large value of γ , sin e P and Q are independent of gain γ and ρ de reases with γ .

Numeri al veri ation The aim of this se tion is to illustrate the performan e of the estimation s heme (11-14) by numeri al simulations using a more realisti AD model than (1) where two degradation stages (a idogeni and methanogeni degradation) for the soluble substrate but also u tuations in the inlet omposition of the digester are onsidered. The main idea here is to verify, via numeri al simulations, if the proposed estimation s heme based on a very simple mathemati al model is able to a hieve the on-line estimation of the VFA's on entration. For this reason we use the AD model developed by Bernard et al. 2001 45 that in ludes inhibition of the methanogeni growth rate. The underlying model assumes two main ba teria populations, the rst one, alled a idogeni ba teria X1 , onsumes organi substrate S1 (total soluble Chemi al Oxygen Demand COD ex ept Volatile Fatty A ids VFA) and produ es VFA, that is onsidered as se ondary substrate S2 through an a idogenesis stage. The se ond population alled methanogeni ba teria X2 , uses VFA as substrate in a methanogenesis stage for growth and produ es methane and arbon dioxide. Thus, the global anaerobi pro ess an be written as the redu ed bio hemi al rea tion network k 1 S1 k 3 S2

µ1 (·)X1

# X 1 + k 2 S2

µ2 (·)X2

#

X2 + k4 CH4

15


(26) (27)


Page 16 of 30

The orresponding mass balan e for a ontinuous AD is given by: S˙ 1 = D(S1f − S1 ) − k1 µ1 (S1 )X1 X˙ 1 = µ1 (S1 )X1 − αDX1 S˙ 2 = D(S2f − S2 ) + k2 µ1 (S1 )X1 − k3 µ2 (S2 )X2

(28)

X˙ 2 = µ2 (S2 )X2 − αDX2

Similarly, in 45 the outow rate of methane gas Qm proportional to the rea tion rate of the methanogenesis stage Qm = k4 µ2 (S2 )X2

(29)

where k4 is the yield oe ient for the methane produ tion. With respe t to the spe i growth rates for the a idogeni and methanogeni populations, in 45 are assumed to be des ribed by the Monod and Haldane expressions, respe tively, i.e., µ1,max S1 KS1 + S1

(30)

µ2,max S2 KS2 + S2 + S22 /KI2

(31)

µ1 (S1 ) =

µ2 (S2 ) =

where µ1,max , KS1 , µ2,max , KS2 , and KI2 are the maximum ba teria growth rate and the halfsaturation onstant asso iated to the substrate S1 , the maximum ba teria growth rate in the absen e of inhibition, and the saturation and inhibition onstants asso iated to substrate S2 , respe tively. In the numeri al simulations nominal parameter values reported in 45 have been used (see Table 1). It is important to remark the following: the estimation s heme (13-14) use only a nominal ve tor eld (5) that is des ribed by a single degradation stage with onstant inlet

omposition. The parameter values used for the nominal ve tor eld (5) are set as follows kt = k3 , km = k4 and µ(S) = µ2 (S2 ) with S = S2 . The observer parameters are set to the

16


Page 17 of 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 1: Nominal parameter values used in the numeri al simulations 45. Parameter

Value

Units

k1 k2 k3 k4

42.14 116.5 268 453 0.05 0.031 7.1 9.28 16 1.0 10 80

g/g mmol/g mmol/g mmol/g h−1 h−1 g/L mmol/L mmol/L −− g/L mmol/L

µ1max µ2max KS1 KS2 KI2 α S1f S2f

following values: γ = 17, κ1 = 17, κ2 = 12, λ1 = λ2 = 1 and q = 3/2. The performan e of the proposed estimation methodology is shown in Fig. 2. In numeri al experiments the omposition of digester input ow is disturbed as is an see in Fig. 1. Also, it is onsidered a hanges in the operating onditions of the AD pro ess are onsidered through hanges in the dilution rate values performed at the time instants t = 500 h and t = 800 h.(Fig. 2). As we an see in Fig. 2, the observer s heme is able to estimate the a tual

value of VFA on entration despite the following: load disturban es in the inlet omposition (Fig. 1); hanges in the operating onditions and un-modeled dynami s. Performan e omparison between the proposed estimation s heme (13-14) and the observer published in 29 are shown in Figures 3-6. For the operation prole of Fig. 3 the estimation value of the observer s heme (13-14) is loser to the a tual VFA on entration. Simulations for dierent operating proles of the AD pro ess are shown in Fig. 4. For these onditions the performan e of the proposed estimation s heme looks better than the performan e of the observer published in. 29 However, ex essive u tuations and peaks for the estimated value of the observer s heme (13-14) are present. In order to have a more pre ise omparison between the performan e of both estimation s hemes a performan e index based on the absolute integral

17



Page 18 of 30

of the estimation error is dened as PI =

Z

tf

|e(t)|dt

(32)

0

where e(t) = Sˆ2 − S2 and tf is the end time of the simulation. A high value of the absolute integral of the estimation error is related with a bad observer performan e, i.e. the lower the value the better the performan e. In Fig. 5 we an see the performan e of ea h observer for the two simulated operation proles. For the rst operating prole shown in Fig. 3 the absolute integral for the proposed observer is of order of the half of the orresponding for the observer. 29 In ontrast, for the operating prole shown in Fig. 4 the value of the absolute integral for the observer 29 is more or less the same. The above is related with the ex essive u tuations for the estimated value of the observer s heme (13-14). However, the sum of the absolute integrals for both operation proles is less for the proposed observer (13-14) than the observer published in. 29 Finally, in Fig. 6 is shown the estimation error for both observers for dierent initial onditions. As we an see in Fig. 6 the initial ondition does not present important ee t on the observers performan e. In the ontext of an experimental implementation of the estimation methodology proposed here it is need a more exhaustive analysis of the observer onvergen e under highly un ertain onditions. Also, it is desirable developed a more pre ise tuning methodology for the observer parameters. Both issues will be addressed in future ontribution.

Con lusion In this paper we proposed a robust software sensor for online estimation of Volatile Fatty A ids (VFA) on entration, as a key variable in Anaerobi Digestion pro ess only from methane outow rate measurements. The observer is based on a lo al oordinate transformation with a super-twisting algorithm that provide robustness properties under disturban es and un ertainty onditions. By numeri al simulations it is shown that the on-line estimation 18


Page 19 of 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


methodology proposed is able to reje t un-modeled dynami s and load disturban es in the AD pro ess.

A knowledgement This work was nan ially supported by the Se torial Fund CONACYT-SENER-Energeti Sustainability from Mexi o through the proje t "Development of monitoring and ontrol

systems to maximize biogas produ tion in wastewater treatment pro esses", (grant 279043).

Referen es (1) Weiland, P. Biogas produ tion: urrent state and perspe tives, Applied Mi robiology and

Biote hnology, 2010, 85, 849-860. (2) Kravaris, C. & Savoglidis, G. Tra king the singular ar of a ontinuous biorea tor using sliding mode ontrol. Journal of The Franklin Institute, 2012, 349, 1583-1601. (3) Lara-Cisneros, G., Aguilar-López, R. & Femat, R. On the dynami optimization of methane produ tion in anaerobi digestion via extremum-seeking ontrol approa h. Com-

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and Te hnology, 2012, 66(5), 1088-1095. (15) Madsen, M., Holm-Nielsen, J.B. & Esbensen, K.H. Monitoring of anaerobi digestion pro esses: A review perspe tive. Renewable and Sustainable Energy Reviews, 2011, 15, 3141-3155. (16) Luenberger, D. An introdu tion to observers. IEEE Transa tions on Automati Control,

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(26) Kal hev, B., Simeonov, I. & Christov, N. Kalman lter design for a se ond-order model of anaerobi digestion. International Journal Bioautomotion, 2011, 15(2), 85-100. (27) Ro ha-Cózatl, E., Sbar iog, M., Dewasme, L., Moreno, J.A. & Vande Wouwer, A. State and Input Estimation of an Anaerobi Digestion Rea tor using a Continuous-dis rete Unknown Input Observer. IFAC-PapersOnLine, 2015, 48(8), 129-134. (28) Rodríguez, A., Quiroz, G., Femat, R., Méndez-A osta, H. & de León, J. An adaptive observer for operation monitoring of anaerobi digestion wastewater treatment. Chemi al

Engineering Journal, 2015, 269, 186-193. (29) Lara-Cisneros, G., Aguilar-López, R., Do hain, D. & Femat, R. On-line estimation of VFA on entration in anaerobi digestion via methane outow rate measurements.

Computers and Chemi al Engineering, 2016, 94, 250-256. (30) Carlos-Hernández, S., Sán hez, E.N. & Béteau, J.B. Fuzzy ontrol stru ture for an anaerobi uidised bed. International Journal of Control, 2012, 85(12), 1898-1912. (31) Aguilar-López, R. & Maya-Yes as, R. State estimation for nonlinear systems under un ertainties: A lass of sliding-mode observer. Jornal of Pro ess Control, 2005, 15, 363-370. (32) Moreno. J. & Do hain, D. Global observability and dete tability analysis of un ertain rea tion systems and observers design. International Journal of Control, 2008, 81, 10621070. (33) Moreno, J. On dis ontinuous observers for se ond order systems: Properties, analysis and desing. In B. Bandyopadhyay et al.(Ed.): Advan es in Sliding Model Control, 2013, LNCIS 440, pp. 243-265. (34) Apaza-Perez, W.A., Fridman, L. & Moreno, J.A. Higher order sliding-mode observers

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Inlet composition

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

100 95 90 85 80 75 70 65 60

10

S

1f

S

2f

0

0

200

400

600

800

1000

1200

time [h]

Figure 1: Load disturban es in the inlet omposition, with S1f [gL−1 ] and S2f [mmolL−1 ], dashed line represent the nominal values.

24


Page 24 of 30

Page 25 of 30

50 Actual value -1

VFA's concentration, [mmol L ]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Estimated value

40

30

20

10

0

0

200

400

600

800

1000

1200

time, [h]

Figure 2: Performan e of the robust observer (12-14) for a four-dimensional AD model (28) with load disturban es in the inlet omposition and hange in the operating ondition at 500 and 800 hours.

25



Actual value

30

Observer from [29]

-1


35

Proposed observer

25 20 15 10 5 0

0

200

400

600

800

1000

1200

1000

1200

Time, [h]

(a)

10 8 6

-1

Estimation error, [mmol L ]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

4 2 0 -2 -4 -6

Observer from [29] Proposed observer

-8 -10 0

200

400

600

800

Time, [h]

(b)

Figure 3: Comparison of the performan e of the proposed observer with the observer published in. 29 26 ACS Paragon Plus Environment

Page 26 of 30

Page 27 of 30

30

-1


Actual value Observer from [29]

25

Proposed observer

20 15 10 5 0

0

200

400

600

800

1000

1200

1000

1200

Tiem, [h]

(a)

10 8

Observer from [29] Proposed observer

6

-1


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4 2 0 -2 -4 -6 -8 -10 0

200

400

600

800

Time, [h]

(b)

Figure 4: Comparison of the performan e of the proposed observer with the observer published in 29 with other operation prole. 27 ACS Paragon Plus Environment


9000

Observer from [29] 7500

Proposed observer

6000

PI (32)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

4500

3000

1500

0 Operating profile 1

Operating profile 2

Sum

Figure 5: Performan e index based on the absolute integral of the estimation error (32) for two operating proles.

28


Page 28 of 30

Page 29 of 30

10 8 6

-1


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4 2 0 -2 -4

Observer from [29] proposed observer

-6 -8 -10 0

50

100

150

200

250

300

350

400

450

500

Time, [h]

Figure 6: Estimation error of the proposed observer and the observer published in 29 for dierent initial onditions.

29



3 5

A c tu a l v a lu e O b s e rv e r fro m [2 8 ] P ro p o s e d o b s e rv e r

-1

]

3 0

V F A ’s c o n c e n t r a t io n , [ m m o l L

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Page 30 of 30

2 5

2 0

1 5

1 0

5

0 0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

T im e , [h ]


1 2 0 0

Software Sensor for Online Estimation of the VFA's Concentration in

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