Ci a ion: Co és, L.G.; Ba bancho, J.;
La ios, D.F.; Ma in-Ba is a, J.D.;
Mohedano, A.F.; Po illa, C.; de la
Rubia, M.A. Full-Scale Diges e s:
Model P edic i e Con ol wi h
Online Kine ic Pa ame e
Iden i ica ion S a egy. Ene gies 2022,
15, 8594. h ps://doi.o g/10.3390/
en15228594
Academic Edi o : Gio anni Esposi o
Recei ed: 21 Augus 2022
Accep ed: 10 No embe 2022
Published: 16 No embe 2022
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Copy igh : © 2022 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
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A ibu ion (CC BY) license (h ps://
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ene gies
A icle
Full-Scale Diges e s: Model P edic i e Con ol wi h Online
Kine ic Pa ame e Iden i ica ion S a egy
Luis G. Co és 1,*, J. Ba bancho 1, D. F. La ios 1, J. D. Ma in-Ba is a 2,3, A. F. Mohedano 3, C. Po illa 4
and M. A. de la Rubia 3
1Depa amen o de Tecnología Elec ónica, Escuela Poli écnica, Uni e sidad de Se illa, 41011 Se ille, Spain
2E uels Technologies L d., 42-44 Bishopga e, London EC2N 4AH, UK
3Depa amen o de Ingenie ía Química, Campus de Can oblanco, Uni e sidad Au onoma de Mad id,
28049 Mad id, Spain
4Facul ad de Minas, Uni e sidad Nacional de Colombia, Robledo, Medellín 050034, Colombia
*Co espondence: luico [email p o ec ed]
Abs ac :
This wo k p esen s a nonlinea model p edic i e con ol scheme wi h a no el s uc u e
o obse e s aiming o c ea e a me hodology ha allows easible implemen a ions in indus ial
anae obic eac o s. In his way, a new s ep-by-s ep p ocedu e scheme has been p oposed and es ed
by sol ing wo speci ic d awbacks epo ed in he li e a u e esponsible o he ine iciencies o hose
sys ems in eal en i onmen s. Fi s ly, he implemen a ion o con ol s uc u es based on modeling
depends on mic oo ganisms’ concen a ion measu emen s; he echnology ha achie es his is no
cos -e ec i e no iable. Secondly, he eac ion a es canno be conside ed s a ic because, in he
ex ended anae obic diges ion model (EAM2), he la ge luc ua ion o pa ame e s is una oidable. To
ace hese wo d awbacks, he concen a ion o acidogens and me hanogens, and he alues o he wo
eac ion a es conside ed ha e been es ima ed by a s uc u e o wo obse e s using da a collec ed by
senso s. A e 90 days o ope a ion, he e o in con e gence was lowe han 5% o bo h obse e s.
Fou model p edic i e con olle (MPC) con igu a ions a e used o es all he p e ious in o ma ion
ying o maximize he olume o me hane and demons a e a sa is ac o y ope a ion in a wide ange
o scena ios. The esul s demons a e an inc ease in e iciency, anging om 17.4% o 24.4%, using as
a e e ence an open loop con igu a ion. Finally, he ope a ional obus ness o he MPC is compa ed
wi h simula ions pe o med by adi ional al e na i es used in indus y, he p opo ional-in eg al-
de i a i e (PID) con olle s, whe e some simple ope a ional scena ios o manage o an MPC a e
longe su icien o dis up a no mal ope a ion in a PID con olle . Fo his con olle , he simula ion
shows an e o close o he 100% o he e e ence alue.
Keywo ds:
anae obic diges ion; asymp o ic obse e ; homogeneous eac ion sys ems; kine ic
pa ame e obse e ; model p edic i e con ol; s ep-ahead
1. In oduc ion
The main a ibu e o indus ial p ocesses and manu ac u ing is he la ge-scale ans-
o ma ion o mass and ene gy in o high-end aluable speci ic ma e ials and compounds [
1
].
In essence, chemical and biological p ocesses a e impe a i e o hose pu poses because i
is p ima ily a his s age whe e he elemen s in a eac ion can be manipula ed o achie e
he desi ed a ibu es [
2
–
4
]. All he abo e is possible i a con ol and moni o ing scheme
is added o gua an ee sa e and s able ope a ion [
5
]. Speci ically, in anae obic sys ems,
he ans o ma ion o subs a es in o biogas and diges a e has a ac i e ad an ages in
compa ison o classical al e na i es such as ae obic diges ion o compos ing [
6
,
7
]; i e u ns
li le sludges, has a posi i e o e all ene gy balance, and also has eno mous po en ial o
educe challenging and concen a ed subs a es such as animal was es [
8
,
9
]. Howe e , his
is an elabo a ed biochemical p ocess ha includes di e se mic oo ganisms, making he
Ene gies 2022,15, 8594. h ps://doi.o g/10.3390/en15228594 h ps://www.mdpi.com/jou nal/ene gies
Ene gies 2022,15, 8594 2 o 23
p ocess nonlinea [
10
]. In his a ea, i is g ea o in e es o de i e models ha would be
as sensi i e as possible o he lack o phenomenological knowledge [
11
,
12
]. Thus, an in-
e es ing al e na i e o ep esen his phenomenon is he ex ended anae obic diges ion
ma hema ical model 2 (EAM2) [
13
], designed o con ol and moni o ing pu poses. I
ci cum en s he di icul ies in loca ing he biological lack o knowledge in dedica ed e ms,
namely eac ion a es [
14
]. Addi ionally, compa ed wi h i s p edecesso , in he anae obic
diges ion ma hema ical model 2 (AM2), wo addi ionally yield pa ame e s we e included
o conside he in luence o p o eins and aminoacids on he dynamic o alkalini y and
conside mo e subs a es o be modeled.
In o de o alloca e a s a egy ha be e i s he challenge men ioned abo e, se e al
echniques ha e been p oposed om di e en con ol a eas. Some o hem a e nonlinea
p opo ional-in eg al-de i a i e (PID) con olle s, linea quad a ic egula ion (LQR) o
nonlinea sys ems [
15
], nonlinea model p edic i e con ol (NMPC) [
16
], and eceding
ho izon con olle s [
17
,
18
], o name he mos ele an cases epo ed in he li e a u e. Al-
hough hese s a egies ha e p o en o be e ec i e, he emaining ine iciencies su ge om
he ac ha some o he pa ame e s a e no a ailable o be measu ed online, and he concen-
a ions o he mic oo ganisms canno be measu ed pe iodically because on-si e analy ical
p ocedu es a e challenging o pe o m [
5
]. A nonlinea model-based con ol scheme has
been p oposed o achie e a sa is ac o y ope a ion. A he same ime, he ep esen a ion o
anae obic diges ion (AD) p ocesses is educed o a minimal numbe o assump ions in he
model-building exe cise, main aining he pe o mance [
18
,
19
]. Among he al e na i es eg-
is e ed in he li e a u e, he Model P edic i e Con ol (MPC) [
17
,
18
] o e s se e al bene i s
compa ed wi h adi ional me hods [
14
,
20
,
21
]. This algo i hm uses a ma hema ical model
o p edic and ollow he sys em’s e olu ion in ad ance. One o he main ad an ages is he
explici use o physical and ope a ional cons ain s in he con olle , allowing i o s and
a ound wo king unde easible ope a ional condi ions [22,23].
To achie e his objec i e, a necessa y condi ion mus be sa is ied o unlock he doo
and enable he ope a ion o con ol schemes based on modeling. All he in o ma ion needed
om he anae obic eac o mus be a ailable pe iodically, so a g oup o eliable senso s
should be conside ed o measu e, in eal- ime, he e olu ion o eac ions in he homoge-
neous eac ion sys em [
24
]. Some key a iables, he concen a ions o mic oo ganisms,
as well as he g ow h a es o eac ions (conside ed a iable due o he la ge luc ua ion o
pa ame e s in AM2 [
10
]), a e usually es ed using analy ical me hods h ough specialized
labo a o ies on-si e. The cos and du a ion o hese analyses limi he equency o he
sampling [
25
]. Addi ionally, he a ailable online senso s on he ma ke o biomass o
me aboli es need o be adequa e o egula indus ial applica ions [
14
]. The p e ious
assump ion se s a signi ican challenge because collec ing his in o ma ion is c i ical o he
ope a ion o he con olle algo i hm. In gene al, he una ailabili y o specialized ins u-
men s incen i e he design and cons uc ion o obse e s dedica ed o he eac ion p ocess.
Hence, he design o so wa e senso s, based on he a ailable measu emen s [
14
,
26
], allows
o gua an ee a s eamline o in o ma ion be ween he eac o and he con ol algo i hm [
27
].
S a e and pa ame e obse e s es ablished o e accu a e and s ic models a e he
bes al e na i es o ind a way o ob ain he lack o he da a needed [
28
–
30
]. Fo anae -
obic eac ion p ocesses, he nonlinea AM2 has been widely p o en o moni o ing and
con ol pu poses [
10
,
13
]. This model, cons uc ed unde a mass-balance amewo k, can
adap i s dynamics o a ious AD con igu a ions o eac o s and elude he d awback
ela ed o he absence o phenomenological knowledge, concen a ing his in o ma ion
in non-sensi i eness-dedica ed e ms named eac ion a es [
10
,
23
]. The complexi y o
he biochemical p ocess conside s di e en ypes o mic oo ganisms, making he p ocess
nonlinea and unp edic able, imp ac ical o he adi ional online es ima ion s a egies,
especially i he ope a ion o he homogeneous sys em is condi ioned by cons ain s [
5
].
To deal wi h hese incon eniences, a sequen ial s uc u e in cascade has been p oposed [
14
].
The main idea is o es ima e he concen a ion o acidogens and me hanogens in he absence
o eac ion a es. Then, he nex s ep is o use his in o ma ion o pe o m a new obse a ion
Ene gies 2022,15, 8594 3 o 23
ha es ima es he eac ion a es. Acqui ing in-dep h knowledge o he e olu ion o eac ion
a es ep esen s a challenge since i e eals he deg ee o comple ion o bo h eac ions
du ing he p ocess [5,31].
This pape is o ganized as ollows. Sec ion 2explains he ex ended AM2 ma hema ical
model used o con ol pu poses, whe e addi ional e ms a e included due o he in luence
o p o eins and aminoacids on he dynamics o alkalini y. The modi ica ion o he AM2
model [
10
] unlocks he doo o add a b oad spec um o subs a es allowed o be used
by he con ol algo i hm. Then, Sec ion 3p esen s he pa ame e iden i ica ion p ocedu e
based on op imiza ion ha uses a no el s ep-ahead algo i hm o conside ably imp o e
he e iciency o adjus ing he dynamics o he model o he da a. The asymp o ic s a e
obse e ha ope a es wi hou in o ma ion on he eac ion a es is p esen ed o es ima e
he concen a ions o acidogens and me hanogens. Nex , using he p e ious in o ma ion,
a second obse e es ima es he e olu ion o he wo eac ion a es conside ed. In Sec ion 4,
he in o ma ion aken om obse e s and he eac o ’s measu emen s is used o enable he
ope a ion o he MPC. Finally, Sec ion 5shows he imp o emen achie ed by using he MPC
con olle , measu ing he inc ease in he olume o me hane p oduced. The obus ness o
he MPC p oposed is compa ed wi h he PID con olle .
2. Op imal Pa ame e Iden i ica ion Algo i hm
The model EAM2, ini ially p oposed by Be na d e al. [
10
] as AM2 in 2002 and modi ied
by Co és e al. [
13
] in 2022, was selec ed o suppo he new moni o ing and con ol
s uc u e p esen ed in his pape . The new model sol es wo issues documen ed in he
li e a u e as he main d awbacks in implemen ing au oma ed decision suppo sys ems on
anae obic eac o s [
11
]. Fi s , he necessi y o cha ac e ize he inle subs a e [
10
]. EAM2
conside s a wide ange o subs a es han AM2 because i akes in o accoun he e ec o
ammonium p oduced om p o eins and aminoacids ha a ec he dynamics o alkalini y.
Second, he concen a ions o acidogens and me hanogens a e es ima ed using a speci ic
s a e obse e . The abo e-men ioned solu ions allow balancing he absence o eliable
senso s o analy ical me hods o ob ain he needed in o ma ion pe iodically [10].
2.1. P oposed Mass-Balance Ma hema ical Model o Anae obic Reac o s
The ex ended ma hema ical model used in his pape is based on he AD ep esen ed
by wo main g oups o mic oo ganisms. Two consecu i e p ocess desc ibe he AD phe-
nomenon on eac o s, he acidogenic bac e ia ( ep esen ed by X
1
) ha consume he o ganic
subs a e ( ep esen ed by S
1
) and p oduce mainly ola ile a y acids (mainly acid ace ic,
ep esen ed by S
2
). Thus, he second g oup o mic oo ganisms, he me hanogenic a chaea
( ep esen ed by X2), consumes S2and p oduces CO2and CH4.
Two yield pa ame e s we e added o ep esen he gene a ion o ammonium due
o he e men a ion o aminoacids o he consump ion by he g ow h o mic oo ganisms.
The e ms added a e K
Z,1
and K
Z,2
, espec i ely. The model p esen ed in Co és e al. [
13
]
conside s he ollowing dynamical equa ions:
dX1
d =X1(µ1−αD)(1)
dX2
d =X2(µ2−αD)(2)
dS1
d =D(S1in −S1)−k1ψ1S1
KS1+S1(3)
dS2
d =D(S2in −S2) + k2ψ1S1
KS1+S1−k3ψ2S2
KS2+S2(4)
Ene gies 2022,15, 8594 4 o 23
dZ0
d =D(Zin −Z) + kZ,1ψ1S1
KS1+S1+kZ,2ψ2S2
KS2+S2(5)
dC
d =D(Cin −C)−qC+k4µ1X1+k5µ2X2(6)
whe e X
1
is he concen a ion o acidogens (kg/m
3
), X
2
is he concen a ion o me hanogens
(kg/m
3
), S
1
is he concen a ion o o ganic subs a e cha ac e ized by chemical oxygen
demand (COD) (kg/m
3
), S
2
is he concen a ion o ola ile a y acids (VFA) (g ace ic
acid/L), Zis he o al alkalini y (kg/m
3
), and Cis he concen a ion o o al ino ganic ca bon
(kg/m
3
). The a iables S
1in
,S
2in
,C
in
, and
Zin
a e, espec i ely, he in luen concen a ions
o S
1
,S
2
,C, and Z. The a iable
α
is ela ed o he eac o design, wi h alues anging
be ween 0 and 1. The alue 0 co esponds o an ideal ixed-bed eac o , and
α=
1, o an
ideal con inuously s i ed ank eac o (CSTR). Dis he dilu ion a e (d
−1
) and is he
in e se o he hyd aulic e en ion ime, HRT = 1/D,
ψ1
is he maximum acidogenic bac e ia
g owing a e (d
−1
),
ψ2
is he maximum me hanogenic a chaea g owing a e (d
−1
), K
S1
is he
hal -sa u a ion cons an o S
1
(kg/m
3
), and K
S2
is he hal -sa u a ion cons an o S
2
(g ace ic
acid/L). The yield pa ame e k
1
is a cons an o subs a e deg ada ion, k
2
is a cons an
o VFA p oduc ion (mmol/kg), k
3
is a cons an o VFA consump ion (mmol/kg), k
4
is a
cons an o CO
2
p oduc ion (mmol/kg),
k5
is a cons an o CO
2
p oduc ion (mmol/kg),
and k
6
is a cons an o CH
4
p oduc ion (mmol/kg). The Monod- ype Equa ions
(7)
and
(8)
a e cha ac e ized by he ollowing wo eac ion a es:
µ1=ψ1S1
KS1+S1(7)
and,
µ2=ψ2S2
KS2+S2(8)
Bo hMonod- ypekine icsdesc ibe heg ow h o acidogenic bac e ia
ψ1
andme hanogenic
a chaea
ψ2
because, du ing he e men a ion p ocess, he biomass does no egis e possible
VFA accumula ion and consequen ly inhibi ion. Finally, he me hane low a e p oduced,
qM, is p opo ional o he eac ion a e o me hanogenesis, as shown in Equa ion (9):
qM=k6ψ2S2
KS2+S2(9)
2.2. Expe imen al Da a Resul s
The da a conside ed o modeling pu poses [
32
] we e used p e iously in Co és e al. [
13
]
o cha ac e ize a CSTR pilo he mophilic diges e (150 L) using he AM2 ex ended ma he-
ma ical model. The o al ex ension o he expe imen was 338 days, whe e he e ec o solid
e en ion ime (SRT) and he dynamics o he sys em in a wide ange o ope a ional condi-
ions we e s udied. Howe e , only a speci ic ange o da a was used, whe e he acidogens
and me hanogens ope a ed unde s able and homogeneous condi ions disca ding uns able
scena ios un a o able o modeling pu poses (see he condi ions a gued o disca d he
da a). Finally, he da a conside ed ep esen a o al o 207 days. The alues o COD a he
inle and VFA and COD a he e luen a e shown in Co és e al. [13].
3. Op imal Pa ame e Iden i ica ion and Online Measu emen s
3.1. Pa ame e Iden i ica ion Using Pa e n Sea ch by S ep-Ahead
E e y sampling ime, he pa ame e iden i ica ion algo i hm s ep-ahead uses he
EAM2 o calcula e he ollowing alue o he sys em in ad ance. The p ocedu e es ablished
ha on he ollowing s ep, he p edic ed alue is compa ed wi h he same a iable measu ed.
The di e ence in magni ude o bo h ep esen s he cumula i e e o . The p ocedu e is
pe o med du ing all he expe imen . The ollowing ec o Es o es all his in o ma ion.
Ene gies 2022,15, 8594 5 o 23
E= [e o 1,e o 2, . . . , e o n](10)
n ep esen s he o al numbe o days e alua ed. Finally, an op imiza ion p oblem is sol ed
aiming o minimize all he e o s calcula ed along he simula ion.
min
p(k)E
subjec o:
nm(k+1) = (nm(k),u(k)), (11)
nnm(k+1) = (nnm(k),u(k)),
0⩽p(k)⩽pmax,∀k=1, . . . ,
nm
ep esen s all he s a es alues measu ed S
1
,S
2
,C, and Z.
nnm
ep esen s all he
non-measu able s a es X
1
and X
2
. The pa ame e s conside ed o be iden i ied a e p(k)
e{µ1max
,
µ2max
,
KS1
,
KS2
,
k1
,
k2
,
k3
,
k4
,
k5
,
k6
,
KZ1
,
KZ2
,
Zin}
.
pmax
ep esen s he maximum
alue a o dable by each pa ame e . This algo i hm has al eady been used in a p e ious
wo k wi h he same elemen s, eal da a, and phenomenological models [13].
3.2. Asymp o ic Es ima o When Reac ion Ra es A e Unknown
The absence o in o ma ion due o he lack o eliable senso s o analysis esul s opens
up an oppo uni y o subs i u e he unce ain y by he design o so wa e senso s. In anae -
obic eac o s, his echnology es ima es he concen a ion o acidogens and me hanogens
wi hou he in o ma ion o he eac ion a es [
11
]. I esul s in a pa icula ca ego y o
obse e -denomina ed asymp o ics, because wo condi ions suppo i s ope a ion; he
sys em is s ill no exponen ially obse able, and he eac ion kine ics a e unknown [
14
].
The de ailed in o ma ion o he design is p esen ed in Co és e al. [13].
In he ollowing, we p esen he condi ions ha a ec he design o he algo i hm:
he in o ma ion o he ma ix
φ
is unknown, he yield coe icien s om
K
a e ully known,
and he numbe o q
s a e
, he numbe o measu ed s a e a iables, is he same o highe han
he ank o he ma ix K( ha is, q
s a e
= dim(
ξ1)≥
ank(K)). Hence, based on he p e ious
in o ma ion, he gene al equa ion o homogeneous eac ion sys ems desc ibed by a gene al
nonlinea s a e space model is p esen ed:
dξ
d =Kφ(ξ, )−Dξ−Q(ξ) + F(12)
whe e dim(
ξ
) = dim(F) = dim(Q) = N, dim(
φ
) = M and dim(K) = N
×
M. Thus, he gene al
nonlinea model Equa ion (12) can be di ided as
dξa
d =Kaφ(ξa,ξb)−Dξa−Qa+Fa(13)
dξb
d =Kbφ(ξa,ξb)−Dξb−Qb+Fb(14)
whe e he ank o Kis p. The subma ix
Ka
esul s om a sec ion o Kwi h p
×
M.
The subma ix
Kb
has he emaining in o ma ion o K. Finally, he ma ices
(ξa
,
ξb)
, (
Qa
,
Qb
),
and (F
a
,F
b)
a e he co esponding pa s o
ξ
,Q, and Fcaused by he in luence o K
a
and K
b
.
The p e ious o mula ion has he ollowing ea u e:
Zob =A0ξa+ξb(15)
The e exis s a ans o ma ion ha conside s
Zob
as a linea combina ion o
ξa
and
ξb
as seen in Equa ion (15). Finally, a e p ocessing Equa ions
(13)
–
(15)
, he new s a e space
model is equi alen o:

Ene gies 2022,15, 8594 6 o 23
dξa
d =Kaφ(ξa,ξb)−Dξa−Qa+Fa(16)
dZob
d =−DZob +A0(Fa−Qa) + (Fb−Qb)(17)
The exp ession
Fa−Qa
= 0 means he pa i ion made by Equa ions
(13)
and
(14)
a e
app op ia e since he new dynamics on
Zob
a e independen om
K
and
φ
. In Equa ion
(16)
,
he independence o
˙
ξa
om
φ
is s a ed. To ex end o de ailed in o ma ion, check he
comple e demons a ion in Co és e al. [13].
Obse e Design
Using he nonlinea EAM2, he ollowing equa ions desc ibe he decoupled subsys em
conduc ed by he s a e a iables X1,X2,S1, and S2:
ξ=



X1
X2
S1
S2



,F=



0
0
DS1in
DS2in



,Q=



0
0
0
0



,K=



1 0
0 1
−k10
k2−k3



,φ=µ1X1
µ2X2(18)
Conside ing he p e ious subsys em, he subsequen s a e equa ion is s uc u ed as
ollows:
d
d 



X1
X2
S1
S2



=



1 0
0 1
−k10
k2−k3



φ1
φ2−D



X1
X2
S1
S2



+



0
0
DS1in
DS2in



(19)
whe e
ξa
and
ξb
g oup he measu able and non-measu able s a es, espec i ely. Thus, using
he equa ions desc ibed in Co és e al. [13], he exp ession o Zob is as ollows:
dZob
d =−DZob +A0(Fa −Qa) + (Fb−Qb)(20)
whe e he ma ices ela ed o he equa ion a e:
ξb=X1
X2,Zob =Zob1
Zob2,A0="1
k10
k2
k1k3
1
k3#,ξa=S1
S2(21)
Fa=DS1in
DS2in,Fb=0
0(22)
wi h:
Zob =Zob1
Zob2=A0ξa+ξb="1
k10
k2
k1k3
1
k3#S1
S2+X1
X2(23)
3.3. Kine ic Pa ame e Reac ion Es ima o
This sec ion p esen s an addi ional ool ha complemen s he lack o measu emen s in
he eac o . The goal is o es ima e he kine ic eac ions coming om he eac ion sys em.
Figu e 1shows he s uc u e p oposed o ob ain all he in o ma ion needed o eed he
ma hema ical model con ained in he con ol scheme s a egy.
Ene gies 2022,15, 8594 7 o 23
Figu e 1. The con olle and he obse e s uc u e p oposed o he anae obic eac o .
The discon inuous line ep esen s he in o ma ion gene a ed by he obse e s. A he
end, he measu ed a iables
nm
and he es ima ed ones con e ge o eed he con olle
algo i hm. In o de o design he kine ic es ima o algo i hm, he ollowing nonlinea
equa ion ha ep esen s he sys em mus be conside ed:
˙
ξ=Kφ−Dφ−Q+F(24)
Equa ion
(24)
assumes ha he coe icien s om
K
a e known, while he dilu ion a e
D
, he inpu lows
F
, and he gas ou pu lows
Q
a e measu ed in eal- ime. Addi ionally, i
is assumed ha
ξ
is ully-known because he non-measu ed dynamics a e econs uc ed by
he p e ious asymp o ic obse e . The ec o
φ
is pa ially known and is di ided as ollows:
φ=Hρ(25)
The ma ix
H
con ains he in o ma ion om he known kine ic eac ions, and
ρ
con-
ains he emaining in o ma ion, he unknown kine ic eac ions. Thus, using Equa ion
(25)
in Equa ion (24) esul s in Equa ion (26).
˙
ξ=KHρ−Dφ−Q+F(26)
The es ima ion o eac ion kine ics,
1
and
2
, is equi alen o es ima ing he en i e
ec o
ρ
. Hence, ollowing he s uc u e p oposed by Bas in e al. [
14
], a new dynamic o
he sys em is p esen ed as ollows:
˙
ˆ
ξ=KH ˆ
ρ−Dξ−Q+F−Ω(ξ−ˆ
ξ), (27)
˙
ˆ
ρ=(KH)TΓ(ξ−ˆ
ξ)(28)
whe e
ˆ
ρ
ep esen s he eal- ime es ima ion o
ρ
. When
ξ−ˆ
ξ∼
=
0 occu s in Equa ion
(28)
, i
means ha
˙
ˆ
ρ∼
=
0 esul s in a pe ec endency p ocess o con e gence o desi able eac ion
kine ics alues
ˆ
ρ∼
=ρ
. The e is a manda o y condi ion. Equa ion
(27)
ha ep esen s he
kine ic eac ion es ima o has an equi alen s uc u e o he Luenbe ge obse e used in
homogeneous eac ion sys ems [14].
˙
ˆ
ξ=Aˆ
ξ+Bu
| {z }
1∗
+Ω(ξ−ˆ
ξ)(29)
Ene gies 2022,15, 8594 8 o 23
The unde lined e m 1
∗
in Equa ion
(29)
,A
ˆ
ξ
+Bu, is equi alen o he e m KH
ˆ
ρ−
D
ξ−
Q
+Fin Equa ion (27). Thus, he abo e-men ioned compa ison esul s in Equa ion (30):
˙
ˆ
ξ=KH ˆ
ρ−Dξ−Q+F
| {z }
1∗
−Ω(ξ−ˆ
ξ)(30)
In his way, Equa ion (30) ep esen s he nonlinea sys em, while Equa ion (29) is he
equi alen linea ized sys em.
Obse e Design
To design he kine ic eac ion es ima o , only he measu ed and es ima ed s a es on
˙
ξ
a e needed, while he e m
ξ−ˆ
ξ
is used as a e e ence o moni o and ollow he es ima o ’s
pe o mance.
ˆ
ξ
should con e ge o
ξ
as soon as possible (
ξ≈ˆ
ξ
). The dynamics o
ˆ
˙
ξ
depends on he e olu ion o he dynamics o he e o
˙
e
= (A
−Ω
L)e. Thus, he e e ences
( he known alues) ha assis in uning he kine ic eac ion es ima o a e
ξ
,
ˆ
ξ
,
K
, and
H
.
Fo be e unde s anding, he ollowing de ini ions a e p oposed: he e m e=
ξ−ˆ
ξ
means
he obse a ion e o , and he e m
˜
ρ=ρ−ˆ
ρ
means he acking e o . Hence, using
Equa ions (31) and (32):
˙
ˆ
ξ=KH ˆ
ρ−Dξ−Q+F−Ω(ξ−ˆ
ξ)(31)
˙
ξ=Kφ−Dξ−Q+F(32)
De i ing on bo h sides o
e
and
˜
ρ
, i esul s in
˙
e=˙
ξ−˙
ˆ
ξ
and
˙
˜
ρ=˙
ρ−˙
ˆ
ρ
. Taking
Equa ions (31) and (32) and eplacing hem by ˙
e esul s in:
˙
e=Kφ−Dξ−Q+F−KH ˆ
ρ+Dξ+Q−F+Ω(ξ−ˆ
ξ)(33)
˙
e=Kφ−KH ˆ
ρ+Ωe(34)
˙
e=KHρ−KH ˆ
ρ+Ωe(35)
˙
e=KH(ρ−ˆ
ρ) + Ωe(36)
˙
e=KH ˜
ρ+Ωe(37)
In he same way, he dynamic o he acking e o is as ollows:
˙
˜
ρ=−(KH)TΓe+dρ
d (38)
Me ging Equa ions (37) and (38) esul s in he ollowing dynamic sys em:
˙
e
˙
˜
ρ="KH ˜
ρ+Ωe
−(KH)TΓe+dρ
d #(39)
Thus, a e o ganizing he p e ious equa ion, i esul s in he ollowing sys em:
˙
e
˙
˜
ρ=ΩKH
−HTKTΓ0e
˜
ρ+"0
dρ
d #(40)
Replacing he ma ices:
A=ΩKH
−HTKTΓ0(41)
V="0
dρ
d #(42)
This esul s in:
˙
e
˙
˜
ρ=Ae
˜
ρ+V(43)
Ene gies 2022,15, 8594 9 o 23
Acco ding o he in o ma ion desc ibed, he es ima ion p ocedu e is pe o med as
ollows:
Ka=−k10
k2−k3,Kb=1 0
0 1,Fa=DS1in
DS2in,Fb=0
0(44)
The nex s ep is o calcula e he ma ix A0; hus, i means ha :
A0=−KbK−1
a(45)
In he ollowing equa ions, he sys em is di ided in o he dynamic s a es o be es i-
ma ed, using he da a om he asymp o ic obse e , Z
1
and Z
2
, and he measu ed da a S
1
and S2. Sol ing ξb om Equa ion (15), we ob ain:
ξb=Zob −A0ξa(46)
Equa ion (46) allows o calcula e he ollowing ma ices:
K=Kb
Ka,H=1 0
0 1,ξ=

ξb
S1
S2
,F=Fb
Fa(47)
Finally, all he in o ma ion needed o supply Equa ions
(27)
and
(28)
a e ob ained abo e.
4. Nonlinea Model P edic i e Con olle (Mpc)
As i is shown in Figu e 1, he obse e s uc u e calcula es he non-measu able
dynamic s a es, he concen a ions o X
1
and X
2
(wi h he asymp o ic obse e ), and he
kine ic eac ion a es
1
and
2
(wi h he eac ion a e obse e ). Once he in o ma ion om
measu emen s and i ual senso s has been deli e ed o he con olle , he algo i hm is
eady o s a calcula ing he con ol ac ions a e ecei ing he es o he in o ma ion: he
ope a ional and physical ins uc ions, as well as he con ol objec i es.
4.1. Con olle Design
The subsequen condi ions eme ge as he equi ed ules o ake in o conside a ion o
design he con olle .
•
The eac o has o be balanced o eac agains pe u ba ions, always wo king wi hin
he physical and ope a ional bounda ies.
• Me hane p oduc ion needs o be maximized all he ime.
•
The en i onmen al egula ions and he capaci y o he eac o o educe he concen-
a ion o subs a es a he inle , S
1
and S
2
, condi ioned he p og amming. The e o e,
he ule S
1
( ) + S
2
( )
≤
K
d
, whe e K
d
deno es he maximum e luen concen a ion o
he bo h subs a es conside ed, has o be ollowed.
•
The eac o has o be p o ec ed agains ailu es due o unexpec ed high a ia ions on
VFA, and in consequence, inhibi ions on he me abolism o mic oo ganisms. Thus,
an al e na i e is o use alkalini y as a base o neu alize he le el o acids.
S2( )
Z( )=λ(48)
Equa ion
(48)
ies o p ese e a ela ion be ween VFA and Zdu ing he ope a ion
o he eac o . Acco ding o he li e a u e,
λ
has o be wi hin 0.1 and 0.3, o main ain he
mic oo ganisms in a zone o com o in ega d o he le el o acids [11].
4.2. S uc u e o he Con olle
Finally, he s uc u e o he con olle is shown in Figu e 2. The dilu ion a e
D
con ols he amoun o subs a e a he inle . I esul s in u
C
, he a iables used by he
MPC o manipula e he dynamics o he eac o . u
i
ep esen s he non-con olle a iables
conside ed as inle : S
1in
,S
2in
,C
in
, and Z
in
.u
pe
holds he in o ma ion abou he inpu s
conside ed as dis u bances. y con ains he in o ma ion o he a iables a ailable o be
measu ed. n
m
a e he s a e dynamics ha can be measu ed: S
1
,S
2
,Z, and C. The alue o pH
Ene gies 2022,15, 8594 16 o 23
5.4. Mpc Con olle wi hou Res ic ions
Figu e 7shows he esul s o maximizing he me hane p oduced by he eac o using
an MPC wi hou he ope a ional and physical es ic ions shown in Equa ion (49).
(a) (b)
(c) (d)
(e) ( )
Figu e 7.
MPC con olle wi hou es ic ions, maximizing he olume o CH
4
. Time cou se o (
a
) ol-
ume o CH
4
; (
b
) dilu ion a e D; (
c
) alue o pH; (
d
) dynamic o VFA; (
e
) alues o
λ
; (
) dynamics o
S2and Z.
Figu e 7a shows a subs an ial inc ease in he olume o me hane p oduced (see discon-
inuous ed line), whe e he black line ac s as he e e ence ( olume o me hane p oduced
due o p o ile inpu s used in pa ame ic op imiza ion), o measu e he imp o emen s on e -
iciency. The co esponden con ol ac ions calcula ed a e shown in Figu e 7b. The absence
o es ic ions in he algo i hm allows he con olle o calcula e con ol ac ions, neglec ing
s ong changes in ope a ion ha could a ec he s abili y o he eac o . As can be seen,
maximizing he objec i e u ns he alues o
D
ex eme; showing oscilla ions om one

Ene gies 2022,15, 8594 17 o 23
ex eme o he o he . Se e al imes he alue o
D
dec eases o ze o and suddenly ise o
he maximum. In he same way, pH had se e e changes e en beyond he ope a ional limi s
(see Figu e 7c). The ela ionship be ween pH and VFA is clea in Figu e 7c,d showing a
co ela ion along he simula ion. This algo i hm exposes high ex e nal isks in ope a ion
because he e exis s a lack o in o ma ion o e he sys em’s ope a ion. Figu e 7e shows
he co esponding alues o
λ
along he simula ion; i appea s ha he alues go beyond
he limi s epea edly. As i seems in Figu e 7 , he alue o alkalini y
Z
exceeds he alue
o
S2
whe e a his poin , he alue o
λ
goes beyond 1, meaning he eac ion sys em
uns inadequa ely.
Gi en he p e ious esul s in Figu e 7, he maximiza ion o me hane wi hou es ic-
ions leads he sys em o ope a e beyond non- easible limi s. The e o e, i is necessa y o
include ex a in o ma ion abou he p ocess o a oid inhibi o s along he simula ion.
Figu e 8shows he esul s ob ained due o he ope a ion o he MPC wi hou he
es ic ion o bu e ing capaci y (
λ
) and wi hou he mul is a unc ion. Figu e 8a shows
he olume o me hane p oduced by bo h he e e ence (black line), and he MPC ( ed
discon inuous line). A signi ican inc ease in he olume o me hane p oduced compa ed
o he e e ence is obse ed. Figu e 8b shows he con ol ac ions calcula ed by he MPC.
As i is shown, he oscilla ion is se e e, howe e , he esul s a e eliable. Figu e 8c,d shows
he esul s o he pH and
S2
. As expec ed, he oscilla ion o alues is in ense a ound he
e e ence (black line) bu no a ac i e o con ol pu poses. Finally, Figu e 8e shows he
e olu ion o he non- es ic ed pa ame e
λ
. As expec ed, due o he absence o es ic ions
on p og amming, he maximum alue o λhas been exceeded once.
(a) (b)
(c) (d)
Figu e 8. Con .
Ene gies 2022,15, 8594 18 o 23
(e) ( )
Figu e 8.
MPC con olle wi hou es ic ions. (
a
) Volume o CH
4
; (
b
) dilu ion a e D; (
c
) alue o pH;
(d) dynamic o VFA; (e) alues o λ; ( ) dynamics o S2and Z.
5.5. Model P edic i e Con olle wi h Res ic ions
The in o ma ion abou ope a ional and physical es ic ions is added o he MPC
aiming o main ain he mic oo ganisms unde easible ope a ional condi ions. As shown in
Figu e 9a, he discon inuous ed line shows he olume o me hane p oduced. The e iciency
is main ained despi e he es ic ions; he e o e, his op ion be e i s he sys em unde
s able condi ions. The co esponden con ol ac ions calcula ed by he op imize (see
Figu e 9b) shows a sligh a enua ion compa ed wi h p e ious simula ions. The esul s
o pH in Figu e 9c show ha he ange o alues becomes close o he e e ence han he
p e ious esul s. Compa ed wi h he p e ious esul s, he acidi ica ion le el is much lowe
(see Figu e 9d). In Figu e 9e, he discon inuous blue line ep esen s he limi s a ed o a
egula ope a ion. As i is shown, he discon inuous ed line ne e su passes he limi s
beyond
λ
= 0.8; in some cases, i ge s close . Howe e , i is because he eac o is p oducing
a high amoun o me hane close o he ope a ional limi s. Figu e 9 shows he e olu ion o
he balance be ween VFA and Z. I is clea ha he bu e capaci y is always unde con ol.
Al hough he algo i hm is es ic ed, he amoun o me hane CH
4
p oduced is highe han
he e e ence.
Finally, Figu e 10 shows simila esul s o hose shown in Figu e 8bu p og ammed
wi hou he mul is a unc ion. Despi e he dec ease in e iciency, he esul s show an
adequa e ope a ion.
(a) (b)
Figu e 9. Con .
Ene gies 2022,15, 8594 19 o 23
(c) (d)
(e) ( )
Figu e 9.
MPC con olle wi h es ic ions, maximizing he olume o CH
4
. (
a
) Volume o CH
4
;
(b) dilu ion a e D; (c) alue o pH; (d) dynamic o VFA; (e) alues o λ; ( ) dynamics o S2and Z.
(a) (b)
(c) (d)
Figu e 10. Con .
Ene gies 2022,15, 8594 20 o 23
(e) ( )
Figu e 10.
MPC con olle wi h es ic ions. (
a
) CH
4
e olu ion; (
b
) D alues calcula ed by op imize ;
(c) pH alues; (d) VFA alues; (e)λo e he p ocess; ( ) e olu ion o S2and Z.
Table 1shows a condensed ision o he p e ious esul s. Two same al e na i es we e
simula ed wi hou he mul is a unc ion. As expec ed, he e is a educ ion in e iciency
because he algo i hm chose a andom ini ial poin o s a he p ocess. The black line
is used as a e e ence o compa e easily he esul s om he me hane p oduced by he
MPC [
13
]. In he case o he MPC con olle ha ope a es wi hou ope a ional es ic ions,
he imp o emen achie ed was 17.4% and 24.4%, espec i ely, o he algo i hm wi h no
mul is a and he mul is a unc ion.
The eac o ha ope a es wi h ope a ional es ic ions shows an imp o emen o 18.8%
and 20.8% o bo h MPC schemes, wi hou and wi h he mul is a unc ion, espec i ely.
Al hough he pe o mance is educed signi ican ly because o he inse ion o es ic ions,
he con inuous ope a ion o he eac o is no b oken and can un pe manen ly.
Table 1. Imp o emen in he me hane p oduced by each MPC con ol scheme.
Schemes Wi hou Res ic ions Wi h Res ic ions
No Mul is a Mul is a No Mul is a Mul is a
Inc ease on e iciency 17.4 % 24.4% 18.8% 20.9%
6. Conclusions
A s uc u e o wo obse e s demons a es o be e ec i ely sol ing wo speci ic d aw-
backs esponsible o he ine iciencies implemen ing con ol schemes based on modeling.
The Asymp o ic obse e ha ope a es in he absence o he eac ion a es demons a es
an e o o con e gence less han 5% es ima ing he concen a ions o acidogens and
me hanogens. Then, he p e ious in o ma ion enables an ups eam ope a ion eeding he
kine ic obse e ha es ima es he eac ion a es wi h an e o o con e gence lowe han
5%. In he ollowing, he es o he pa ame e s, conside ed as s a ic due o hei co ela ion
wi h he speci ic biochemical cha ac e is ics o he o ganic ma e , we e calcula ed using
algo i hms based on op imiza ion. All he abo e-men ioned aspec s ep esen he key ha
makes i possible o ope a e MPCs in he indus y. Fou MPC con igu a ions we e used
o es he obse e and iden i ica ion s uc u e p oposed. The objec i e was o maximize
he olume o me hane. While he con olle wi hou he ope a ional es ic ions showed a
conside able inc ease in e iciency, a e ope a ional limi s we e added o he algo i hm,
he e iciency o he con olle sligh ly dec eased. Howe e , i s educ ion on e iciency
is a o dable because he algo i hm helped he sys em o s ay in ope a ion con inuously.
Wi hou ope a ional es ic ions, he e iciency eached 24.4%, while he es ic ed algo i hm
showed an e iciency o 20.9%. The exhaus i eness in he me hodology p oposed is wo h
using because he added in o ma ion gua an ees obus ness in ope a ion and an inc ease
in e iciency compa ed wi h adi ional con ol al e na i es.
Ene gies 2022,15, 8594 21 o 23
Au ho Con ibu ions:
Concep ualiza ion, L.G.C., J.B., D.F.L., J.D.M.-B., C.P. and M.A.d.l.R.; Da a
cu a ion, L.G.C., C.P. and M.A.d.l.R.; Fo mal analysis, D.F.L., J.D.M.-B., C.P. and M.A.d.l.R.; In es iga-
ion, L.G.C., J.B., D.F.L., J.D.M.-B., A.F.M., C.P. and M.A.d.l.R.; Me hodology, L.G.C., J.B. and C.P.;
P ojec adminis a ion, J.B. and C.P.; Resou ces, J.D.M.-B. and A.F.M.; So wa e, L.G.C.; Supe ision,
D.F.L., J.D.M.-B. and M.A.d.l.R.; Valida ion, D.F.L. and M.A.d.l.R.; Visualiza ion, L.G.C.; W i ing—
o iginal d a , L.G.C.; W i ing— e iew & edi ing, J.B., D.F.L. and C.P. All au ho s ha e ead and
ag eed o he published e sion o he manusc ip .
Funding:
The au ho s wish o exp ess hei g a i ude o Fundación FIDETIA (G91045419), Uni e si-
dad de Se illa, Spain, o unding his wo k. The au ho Luis G. Co és exp ess special acknowledge-
men s o Fundación Cen o de Es udios In e disciplina ios Básicos y Aplicados (CEIBA), Colombia,
o unding. Labo a o io de In es igaciones en Ca álisis y nue os ma e iales LICATUC (Uni e sidad
de Ca agena).
Da a A ailabili y S a emen :
The ini ial expe imen al esul s de eloped in his s udy a e in he nex
link: h ps://doi.o g/10.1016/j.p ocbio.2005.03.073 (accessed on 20 Augus 2022).
Con lic s o In e es :
The au ho s decla e ha hey ha e no known compe ing inancial in e es s o
pe sonal ela ionships ha could ha e appea ed o in luence he wo k epo ed in his pape .
Nomencla u e
C,Cin To al ino ganic ca bon concen a ion (kg/m3)
DDilu ion a e (d−1)
k1Yield subs a e deg ada ion
k2Yield o VFA p oduc ion
k3Yield o VFA consump ion
k4Yield o CO2p oduc ion
k5Yield o CO2p oduc ion
k6Yield o CH4p oduc ion (m3)
KS1Hal -sa u a ion cons an (kg/m3)
KS2Hal -sa u a ion cons an (kg/m3)
KZ,1 Yield o ammonium p oduc ion (kg/m3)
KZ,2 Yield o ammonium p oduc ion (kg/m3)
qCCa bon dioxide low a e (m3/d)
qMMe hane low a e (m3/d)
1, 2Reac ion a es (d−1)
S1,S1in O ganic subs a e concen a ion (kg acid ace ic/m3)
S2,S2in Vola ile a y acids concen a ion (kg/m3)
X1Concen a ion o acidogenic bac e ia (kg/m3)
X2Concen a ion o me hanogenic bac e ia (kg/m3)
Z,Zin To al alkalini y (kg/m3)
αF ac ion o bac e ia in he liquid phase
µ1Speci ic g ow h a e o acidogenic bac e ia (d−1)
µ2Speci ic g ow h a e o me hanogenic bac e ia (d−1)
ψ1Maximum acidogenic bac e ia g ow h a e (d−1)
ψ2Maximum me hanogenic bac e ia g ow h a e (d−1)
nmVec o o measu ed s a e a iables
nnm Vec o o non-measu able s a e a iables
p,pmax Vec o o pa ame e s
u,umin,umax Con ol ac ion (d−1)
ξVec o o s a e a iables
KMa ix wi h he kine ics o he biochemical
φMic obiological eac ions in ol ed on sys em
QGaseous a e o mass ou low om eac o
FMass eed a e due o ex e nal subs a es
ρUnknown unc ions o all s a es
HKnown unc ions o all he s a es
ΓGain ma ix o he upda ing law
ΩSqua e gain con e gence ma ix

Ene gies 2022,15, 8594 22 o 23
λBu e capaci y
NpP edic ion ho izon (d)
NuCon ol ho izon (d)
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