THE
USE
OF
MODELS
IN
FERMENTATION
CONTROL
T.
Yoshida
and
H.
Taguchi
Depa men
o
Fe men a ion
Technology
Facul y
o
Enginee ing,
Osaka
Uni e si y
Yamada-kami,
Sui a-shi,
Osaka
565,
Japan
Abs ac
Recen ly
he e
has
been
a
s ong
in e es
in he
di ec
digi al
con ol
o
e men a ion
p ocesses.
Mo e
e o
on
iden i ica ion
o
modeling
o
he
p ocesses
is
indispensable
o
accomplish
his
highly
sophis ica ed
con ol.
This
pape
was
w i en
o
p esen
a
undamen al
iew
on
model
cons uc ion,
p ocess
iden i ica ion,
pa ame e
es ima ion,
analysis
o
model
and
examples
o
uses
o
model.
Models
in he
pape s
su eyed
in
he
las
i e
yea s
a e
p esen ed
classi ying
hem
in o
h ee
ca ego ies:
subcul u e,
subcellula
and
submolecula
models.
Assessmen
o
model by means
o
sensi i i y
analysis
and
se e al
app oaches
o
modi y models
o
p ocess
con ol
a e
discussed.
94
T.
Yoshida,
H.
Taguchi
1.
In oduc ion
Con inuous
de elopmen s
in
e men a ion
echnology
suppo ed
by
emendous
ad ances
in
molecula
biology
and
o he
ela ed
ields
ha e
esul ed
in
mo e
p ecise
unde -
s anding
o
mic obial
sys ems.
Mo e
quan i a i e
exp essions
o
e men a ion
p ocesses
ha e
accumula ed
as
a
esul
o
p og ess
in
s udies
o
e men a ion
kine ics
by
a ious
kinds
o
app oaches,
some
o
which
ha e
been
de eloped
in
ela ed
ields,
mainly
chemical
enginee ing.
The
scale
o
e men a ion
p oduc ion
is
g owing
and
mo e
so-
phis ica ed
echnology
o
con ol
enginee ing
migh
be
applied
o
ou
ield
in
conse-
quence.
A
compu e
aided
con ol
sys em
is
he
mos
c a ed
aid
o
e men a ion
sys ems,
because
he
e men a ion
sys em
has
a
mul iplici y
o
a iables.
P ocess
con ol
means
no
only
he
p ac ice
o
an
op imal
policy bu
also
sol ing
he
op imum
p oblems. The
iden i ica ion
o
he
model
is
essen ial
o
p ocess
con ol.
Takama su
[1]
epo ed
he
esul s
o an
inquisi ion
abou
he
way
o
aise
he
e iciency
o
a
compu e coupled
sys em.
27 o
83
answe s
om
he
chemical
enginee s
wo king,
o
p ac ical
use
o
a
compu e ,
eques ed
imp o emen
o
de elop-
men
o
he
ma hema ical
model,
and
ano he
17
wan ed
a
mul iple
a iable
con ol
sys em.
The
in es iga ion
done
wi h
chemical
enginee s
may no
ag ee
wi h
one
wi h
biochemical
enginee s.
The
esul
is,
howe e ,
e y
in e es ing
and
he
au ho s
expec ed
a
s onge
eques
o
modeling
o
e men a ion
p ocesses.
Up
o
now
many
models
ha e
been
p esen ed,
analysed
and
applied.
Fi s ,
a
gene al
concep
o
modeling
o
e men a ion
p ocesses,
is
desc ibed
and
examples
o
models
a e
shown
acco ding
o
a
classi ica ion
in o
h ee
ca ego ies.
Some
echnical
me hods
o
p ocess iden i ica ion
and
pa ame e
es ima ion
a e
lis ed.
A e
applica ions
o
sensi i i y
analysis
is
in oduced,
modi ica ion
o
simple
models
and
e inemen
o
complex
models
a e
discussed.
Finally,
a
couple
o
examples
o
he
uses
o
models
in
ac ual
e men a ion
con ol
and
in
de eloping
con ol
algo i hms
a e
shown.
2.
A
Gene al Concep
o
Fe men a ion
Models
The
pe o mance
o
e men a ion p ocesses
is
based
on
he
biological
ac i i y,
which
is
an
exhibi ion
o
in e ac i e
enzyme
ac ion.
Main
o ganisms
employed
o
indus ial
pu poses
a e
mic oo ganisms
such
as
bac e ia,
ungi
and
s ep omyces.
Many
echniques
o
modeling
and
simula ion
de eloped
in
chemical
enginee ing
can
be
applied.
Howe e
mo e
e o s
a e
s ill
necessa y
o he
quan i a i e
unde s anding
o
he
esponse
o
mic oo ganisms
o
en i onmen al
ac o s.
Recen ly
he e
has
been
a
e-
mendous
de elopmen
o
he
ins umen a ion
o
de ec
and
con ol
en i onmen al
ac o s
and
his
ad ance
has
allowed
mo e
de elopmen
o
modeling.
A
g ea
a ie y
o
models ha e
been
de eloped,
he
amous
Monod
exp ession
[2]
o
T.
Yoshida,
H.
Taguchi
95
mic obial
cell
g ow h
is
he
i s
example.
Acco ding
o
he
li e a u es
(e.g.
Tsuchiya
[3])
models
we e
classi ied
in o
wo
ypes,
namely
uns uc u ed
models
in
which
a
uni o mly
dis ibu ed
cell
mass
is
conside ed
along
wi h
he
ela ionship
be ween
he
biomass
and
p oduc ion
and
s uc u ed
models
in
which
he
cell
s uc u e
is
conside ed,
he
o al
ni ogen
con en
is
di e en ia ed
(DNA,
RNA,
p o ein),
and
u he mo e
enzyme
ac i i ies
in
me abolic
pa hways
a e
possibly
aken
in o accoun .
Fe men a ion
sys ems
show
complex
nonlinea
cha ac e is ics
caused
by
se e al
hun-
d ed
in e ac i e
enzyma ic
eac ions.
I
migh
be
impossible
o
desc ibe
he
sys em
in ol ing
all
enzyme
eac ions.
The e o e
one
needs
o
ex ac
a
mas e
ule
om
he
many
obse a ions
on
he
esponses
o
he
sys em
o
en i onmen al
ac o s
o
o
pick
up a
a e-limi ing
one
ou
o
he
many
eac ions
in
he
sys em.
Wi h
his
backg ound
in
mind
we
like
o
di ide
he
models
o
e men a ion
p ocesses
in o
h ee
ca ego ies
in
he
same
way
as
Koga
e
al.
[4]
sugges ed,
namely:
subcul u e,
subcellula
and
submolecula
models.
|
(1)
Subcul u e
model
In
subcul u e
models
he
mic obiological
sys em
is
deal
wi h
a
an
a e aged
mac oscopic
le el
and
many
o
hem
a e
accep ed
as
he
p ocess
model
o
p ac ical
use
in
design
and
con ol
o
e men a ion
p ocesses.
I
is
assumed
ha
he
a e
o
g ow h
and
me abolic
ac i i y
a e
explici
unc ions
only
o
he
s a e
o
he
sys em,
and
no
o
ime
as
ollows,
=
CC)
(1)
whe e
€
is
a
a iable
ec o .
When
he
s a e
in
he
model
e e s
o
popula ion
densi y,
he
g ow h
a e
is
exp essed
as
a
unc ion
o
popula ion
densi y:
us
(x)
(2)
whe e
,
is
he
g ow h
a e,
namely
he
a e
o
p oduc ion
o
cell
mass
pe
uni
olume
o
cul u e
and
X
is
he
popula ion
densi y,
o
he
cell
mass
pe
uni
olume
o
cul u e.
Nyi i
[5]
has
so ed
ou
many
uns uc u ed
models
o
he
g ow h
a e
and
he
p oduc ion
a e.
The
i s
ype
o
he
g ow h
model
shows
exponen ial
g ow h,
=
ux
(3)
x
|
whe e
LU
is
a
kine ic
coe icien
e med
he
speci ic
g ow h
a e,
which
is a
unc ion
:
o
en i onmen al
ac o s
as
discussed
la e .
Khan
and
Ghose
[6]
p esen ed
he
same
|
kind
o
model
ha
Kono
and
Asai
[7,
8]
de eloped.
Thei
model
is
based
on
he
ac
ha
he
alue
o
coe icien
U
changes
acco ding
o
he
phase
in a
ba ch
cul u e,
namely,
induc ion, ansien ,
exponen ial
g ow h
and
declining
g ow h.
The
nex
is
he
logis ic
law
model:
96
T.
Yoshida,
H.
Taguchi
Bam
HX(1
-
BX)
(4)
This
simple
model
seems
o
be
use ul
o
he
simula ion
o
a
ba ch
cul u e
a
cons an
en i onmen al
condi ions.
Cons an inides
ied
o
p edic
he
op imal
empe a u e
p o iles
o
ba ch
e men a ion
o
penicillin
using
he
logis ic
model.
Monod
assumed
he
ollowing
s oichiome ic
ela ion
be ween
g ow h
and
consump ion
o
he
subs a e
which
limi s
g ow h:
E
--+Y
(5)
whe e
s is
he
consump ion
a e
o
he
limi ing
subs a e
and
Y
is
e med
he
yield
cons an .
Nex
Monod
ecognized
ha
he
g ow h
a e
could
ha e
a
maximum
alue,
and
he
pos ula ed
ha
he
subs a e
dependence
o
g ow h
a e
ollowed
he
Michaelis-
Men en
o m;
su e
2
(6)
whe e
S
is
he
concen a ion
o
he
limi ing
subs a e
and
un
is
he
maximum
speci ic
g ow h
a e
ob ained
when
S
is
much
g ea e
han
he
cons an
Ky:
which
is
e med
he
hal - eloci y
cons an .
Eqs.
(5)
and
(6)
oge he
a e
a
comple e
s a emen
o
Monod's
model,
and
hey
may
be
applied
o
a ious
cases,
o
op imiza ion,
p ocess
analysis
and
p ocess
con ol.
In
he
con inuous
e men o ,
u_Ss
dx
m
qe
h
gaa
e ”
s
us
as
_
Lam
(8)
de
an
S R
_
whe e
Sn
is
he
concen a ion
o
limi ing
subs a e
in
he
eed.
Usabili y
o
Monod's
model
o
e men a ion
p ocesses
has
been
in es iga ed
by
many
au ho s.
They
ecognize
he
accep abili y
o
he
model
o
he
analysis
o
s eady
s a e,
hough
wi h
some
modi ica ions,
e.g.
conside a ion
o
endogenous
e m
in
equa ions.
Ex ensi e
appli-
ca ion
o
he
model
o
he
p edic ion
o
ansien
beha iou
has
been
ied.
Many
au ho s
[9-12]
claimed,
as
desc ibed
in
a
la e
i em,
ha
modi ica ions
should
be
made
o
he
model
o
analysis
o
ansien
s a e.
Recen ly,
mul i-subs a e
limi ing
sys ems
we e
subjec ed
o
s udy.
Sinclai
and
Ryde
[13]
de eloped
models
o
con inuous
cul u e
unde
bo h
ca bon
and
oxygen
limi -
ing
condi ions.
I
was
ound
ha
Monod
kine ics
we e
applicable
o
he
g ow h
a e
dependence
on
oxygen
concen a ion
bu
Con oi's
kine ics
we e
supe io
o he
co -
esponding
dependence
on
ca bon
subs a e
concen a ion.
Fujio
e
al.
[14]
also
p esen ed
a
model
o
ed-ba ch
cul u e
unde
ca bon
and
oxygen
limi ing
condi ions.
Spi ze
[15]
analysed
he
sys em
unde
con ol
o
subs a e
and
pH
by
means
o
he
model
which
includes
he
equa ion
o
change
in
concen a ion
o
hyd ogen
ions.
T.
Yoshida,
H.
Taguchi
97
Some
p oduc s
cause
a
educ ion
in
g ow h
a e.
Se e al
ela ionships
ha e
been
de eloped
o
cha ac e ize
his
educ ion,
one
is
an
exponen ial
educ ion
in
g ow h
a e
[16]:
=
UL
exp
(-kP,)
(9)
Eganbe die
e
al.
[17]
conside
ha
a
ec angla
hype bola
i s
he
da a
mo e
co ec ly
bed
me
(10)
Hinshelwood
[18]
has
shown
linea
inhibi ion
o
he
o m
I
w=
u,
G.0-4
P,)
(11)
Zines
[19]
in es iga ed
he
s eady
s a e
and
he
dynamic
p ope ies
o
he
e men a i e
bac e ium,
K.
ae ogenes,
using
he
Hinshelwood
model.
F equency
esponse
analysis,
using
sinusoidal
a ia ions
in
he
dilu ion
a e
showed
ha
e hanol
inc eased
he
ime
cons an
o
he
me abolic
pa ame e .
(2)
Subcellula
model
Mic obial
p oduc s
ha e
been
b oadly
classi ied
in o
wo
g oups,
"g ow h
associa ed"
and
"nong ow h
associa ed".
This
classi ica ion
was made
acco ding
o
whe he
he
p oduc
is
o med
while
he
o ganism
is
in
he
exponen ial
o
in
he
s a iona y
phase.
An ibio ics
(e.g.,
penicillin, no obiocin,
s ep omycin),
oxin,
and
an igens,
which
a e
seconda y
me aboli es,
ha e
been
included
in
he
g oup
o
nong ow h
associa ed
p oduc s.
Models
o
seconda y
me aboli e
p oduc ion
mus
desc ibe
he
a ia ions
o
p ope ies,
in
o he
wo ds,
chemical
composi ion
and
me abolic
ac i i ies
o
indi-
idual
cells.
Sawada
and
Kojima
[20]
u ilized
he
s uc u ed
model
de eloped
by
Tsuchiya
e
al.
[3]
o
he
in es iga ion
o
he
e ec
o
mic omixing
on a
con inuous
ope a ion.
In
he
model,
cells
a e
di ided
in o
wo
kinds,
iable
cells
and
inac i e
cells.
The
la e
we e
iable
cells
be o e
he
p oduc ion
o
an
inhibi o y
subs ance
by
he
cells.
This
model
can
p edic
he
g ow h
cu e
including
lag,
exponen ial
and
declining
g ow h
and
s a iona y
phase
and
he
e ec
o
age
o
innoculum
on
he
g ow h
o
cul u e.
Ano he
example
o
a
subcellula
model
is
based
on he age
app oach
associa ing
he
p oduc
biosyn hesis
a e
wi h
he
age
o
he
cul u e.
Shu
[21]
exp essed
he
a e
o
p oduc ion
as
a
unc ion
o
he
age
o
he
cell
(6).
eg
z
A,
exp
(-k,
9)
(12)
Blanch
and
Roge s
[22]
de eloped
a
model
o
g amicidin
S
acco ding
o
he
concep
ha
each
cell
in
a
cul u e
ollows
a
p ede e mined
gene ic
o de
in
ca ying
ou
i s

98
T.
Yoshida,
H.
Taguchi
me abolic
unc ions.
The
o ma ion
o
p oduc s
by
an
o ganism
can
he e o e
be
con-
side ed
as
a
unc ion
o
cell
age.
The
cell
age
is
a bi a ily
di ided
in o
wo
g oups,
iz.,
"imma u e"
and
"ma u e",
whe e
ma u e
cells
o m
he
p oduc .
B own
and
Vass
[23]
also
p oposed
a
me hod
o
de e mine
he
ma u ing
ime
and
applied
i
o
se e al
an ibio ic
p oduc ions.
Ma sumu a
e
al.
[24]
analysed
cell
beha iou
in
he
ansien
s a e
o
a
con inuous
cul u e
wi h
a
sophis ica ed
"appa en
age"
model
conside ing
he
esidence
ime
dis ibu ion
in a
e men e .
(3)
Submolecula
model
P og ess
in
s udies
on
cell
physiology
and
me abolism
allows
he
cons uc ion
o
mo e
p ecise
models
which
desc ibe
changes
o
chemicals
in
cells.
Pe inge
e
al.
[26]
p esen ed
a
ma hema ical
model
using
he
concep
o
an
equilib ium
be ween
he
ATP
equi emen s
and
he
ATP
p oduced
by
ca abolic
p ocesses.
An
analy ical
exp ession
o
he
ime
cou se
o
g ow h
could
be
de i ed
which
p o ides
a
quan i a i e exp ession
o
he
Pas eu
e ec ,
p edic s
he
alue
o
Y
ins an aneous
alues
o
P/O
a ios,
ATP?
he
quan i y
o
subs a e
and
oxygen
consumed
and
o
biomass
p oduced.
K eme
[27]
de i ed
a
kine ic
model
exhibi ed
by
ace yl
Co-A
me abolism
and ATP
egene a ion
o
in es iga e
possible
con ol
mechanisms
in
e acycline
p oduc ion.
Monod
[25]
has
de eloped
he
mos
a o ed
hypo hesis
on
induc ion
and
ep ession
o
enzyme
p oduc ion
in
cells.
His
ope on
hypo hesis
is a
splendid
example
o
ecen
ad ances
in
molecula
biology
and
e y
help ul
o
in es iga e
mic obial
beha iou
e en
om
a
mac oscopic
poin
o
iew.
The
hypo hesis
is
cons uc ed
wi h
he
heo ies
o
gene ical
egula ion
o
enzyme
p oduc ion
wi h
chemical
e ec o s,
namely,
ep esso
and
induce .
Models
based
on
he
ope on
hypo hesis
ha e
been
de eloped
o
quan i a i e
exp ession
o
indus ial
e men a ion
p ocesses.
P e e en ial
syn hesis
o
a
enzyme,
a-galac osidase,
was
desc ibed
in
submolecula
model
by
Imanaka
e
al.
[28].
Enzyme
p oduc ion
a e
is
p opo ional
o
he
concen a ion
o
mRNA
in
cells
(M):
dE
eae
keM
-
LE
(13)
whe e
E
deno es
enzyme
con en
in
cells
and
he
second
e m
on
he
igh
hand
side
o
he
equa ion
ep esen s
dilu ion
by
g ow h.
m-RNA
is
syn hesised
wi h
nega i e
co -
ela ion
o
he
con en
o
ep esso
(R)
and
is
subjec
o
i s
o de
decomposi ion:
dM
de
ke
(R,
-
R)
-
k7M
-
uM
(14)
R.
is
he
c i ical
concen a ion
o
ep esso
o
s op
he
syn hesis
o
m-RNA.
The
change
o
ep esso
con en ,
R,
and
a
complex
o
ep esso
and
induce ,
Rön;
>
we e
exp essed
as
ollows:
dR
_
——
ae
denen
ue
eas
kuRS,,
+
ksRS,
-
uR
(15)
T.
Yoshida,
H.
Taguchi
99
BL
a
e
O
Da
HE =
kuRS,
KsBS,.
URS,
(16)
whe e
Sai
is
he
in acellula
concen a ion
o
induce ,
galac ose.
This
model
ga e
a
good
p edic ion
o
he
ba ch
and
con inuous
cul u e
o
a-galac osidase
p oduc ion
[29]
and
was
u ilized
o
he
op imiza ion
o
he
enzyme
p oduc ion
p ocess
[30].
Van
Dedem
[31]
p oposed
he
same
kind
o
model
desc ibing
he
deg ee
o
ep ession
o
induc ion
o
ex acellula
enzyme
syn hesis
as
a
unc ion
o
nu ien
concen a ion.
3.
P ocess
Iden i ica ion
and
Pa ame e
Es ima ion
Iden i ica ion
o
dominan
biological
o
chemical
a e
p ocesses
is
essen ial
o
he
op imal
ope a ion
and
he
bes
pe o mances
o
he
e men a ion
p ocess,
al hough
in o ma ion
abou
mass
and
hea
ans e
and
mixing
cha ac e is ics
is
some imes
necessa y.
The
esponses
o
he
p ocesses
o
he
change
o
ci cums ances
could
be
in es iga ed
in
wo
ways:
expe imen s
wi h
he
ac ual
p ocesses
o
simula ions
o
he
p ocesses
by
means
o
models.
(1)
Simula ion
o
e men a ion
p ocesses
Simula ion
is
de ined
in
he
p esen a icle
as
he
desc ip ion
o
eal
sys ems
o
e men a ion
p ocesses
by
use
o
ma hema ical
equa ions;
compu e
me hods
a e
gene ally
equi ed
o
hei
solu ion.
Blanch
and
Dunn
[32]
p esen ed
a
ull
commen a y
on
modeling
and
simula ion
in
biochemical
enginee ing.
Se e al
impo an
s eps
can
be
ecognized
in
any
simula ion
de elopmen
as
shown
in he
igu e.
Fi s ,
he
pu pose
o
Theo e ical
Backg ound
ee
eee
—
|
Chemical
Kine ics
N
|
Enzymology
|
Ma hema ical]
„|
Pa ame e
Cell
Physiology
Model
Es ima ion
Popula ion
Dynamics
T anspo
Phenomena
A
De e mina ion
o
Expe imen al
Condi ions
S eps
o
simula ion
o
e men a ion
p ocesses
simula ion
mus
be
clea ly
de ined
and
a
good
insigh
in o
eal
e men a ion
phenomena
is
qui e
help ul
o
modeling.
The
second
s ep
is o
w i e
he
equa ions.
The e
a e
wo
s a ing
poin s o
his
s ep:
one
is
heo e ical
analysis
and
he
o he
is
analy-
sis
o
expe imen al
esul s.
Fundamen al
knowledge
abou
e men a ion
and
ela ed
100
T.
Yoshida,
H.
Taguchi
ields
is a
need
o
bo h
cases.
The
hi d
is
he
pa ame e
es ima ion
by
expe imen s
wi h
he
aid
o
ma hema ical
o
nume ical
echniques.
The
expe imen al
condi ions
depend
on
he
me hod
o
pa ame e
es ima ion.
Finally,
he
alidi y
o
he
model
should
be
jus i ied
wi h
expe imen s,
and
econs uc ion
o
model
should
be done
i
necessa y.
Chi and
Howell
[10]
p esen ed
a
ypical
example
o
a e
p ocess
iden i-
ica ion
compa ing
wo
compe ing
models.
They
compa ed
a
modi ied
Powell-Ie usalenskii
bo le
neck
model
o
a
Monod-Haldane
model
o
he
p edic ion
o
he
ansien
beha iou
o
a
con inuous
cul u e.
They
used
Pseudomonas
which
has
a
cha ac e is ic
o
subs a e
inhibi ion.
Applying
Monod-Haldane
model,
us
le
ONE
an
Ss 4;
Following
Powell
[33],
hey
in oduced
Q
subs a e
which
is a
quan i a i e
exp ession
o
he
physiological
s a e
o
he
mic oo ganisms
and
go e ns
he
g ow h
as
ollows:
lL
=
Q
YG
(S)
(18)
and
he
change
o
Q
was
exp essed
in he
ollowing
equa ion:
aQ
_
s
Y
s
d
Um
K
#8487
7K,
a
CR_+5+057/R,
(19)
whe e
C
is
he
a io
o
he
lowe
limi
o
Q,
Q,
o
he
uppe
limi
A
The
obse ed
ansien
beha iou
was
p edic ed
by
he
new
model,
bu
could
no
be
p edic ed
by
a
Monod-Haldane
model.
(2)
Techniques
o
pa ame e
es ima ion
Pa ame e
es ima ion
is
one
o
he
mos
impo an
p oblems
in
he
iden i ica ion
o
p ocesses.
Values
o
pa ame e s
mus
be
es ima ed
by
analysis
o
expe imen al
esul s
o
he
mos
a ional
explana ion
o
he
beha iou
o
he
p ocess.
Obse a ions
o
Measu emen s
ob ained
by
expe imen s
a e
a he
andom
a he
han
de e mis ic,
since
unde
ac ual
condi ions,
p ocess
noise,
cycling,
signal
noise
and
o he
phenomena
in e e e
wi h
all
measu emen s.
Linea
leas
squa es
es ima ion
me hod
has
been
employed
o
pa ame e
es ima ion
o
a e
p ocess
in
e men a ion.
Many
pa ame e
es ima ion
me hods
ha e
been
de eloped
o he
nonlinea
pa ame e
sys em.
The
leas
squa es
pa ame e
es ima ion
does
no
equi e
p io
knowledge
o he
dis ibu ion
o
unobse ed
e o s,
and
i
yields
unbiased
es ima es,
and
esul s
in
he
minimum
a iance
among
all
he
linea
unbiased
es ima ion
me hods,
he
leas
squa es
echnique
is
used
ex ensi ely.
Values
o
pa ame e s
a e
es ima ed
o
minimize
n
o=,2,
[¥-G@lwiy-6]
(20)
whe e
Y
is
he
ma ix
o
obse a ion
and
G
is
he
ma ix
o
p edic ion.
I
Wis
a
diagonal
ma ix,
becomes
a
"weigh ed
leas
squa es"
c i e ion,
i
W
is
a
uni
ma ix
is
he
"o dina y
leas
squa es"
c i e ion.
Box
and
D ape 's c i e ion
[34]
was
applied
by
Johnson
and
Be ho ex
[35]
o
es ima e
wo
pa ame e s
in
a
Monod
model
o
con inuous
T.
Yoshida,
H.
Taguchi
101
cul u e.
They
showed
ha
he
con idence
egion
o
es ima ed
pa ame e s
wi h
bo h
he
da a
o
cell
concen a ion
and
subs a e
concen a ion
is
much
smalle
han
wi h
he
da a
o
any
o
he wo
alone.
Sunds om
e
al.
[36]
s udied
esponse
o
biological
eac o s
o
he
sinusoidal
a ia ions
o
subs a e
concen a ion
and
employed
he
ollowing
c i e ion.
Ss
-
oN
2
xX
-
X,
2
o=2
ica
ee
e
l,
(21)
e
ie
whe e
subsc ip s
p
and
e
deno e
p edic ion
and
expe imen ,
espec i ely.
Bode
diag am
has been
u ilized
o
es ima e
pa ame e s
o
analyse
he
esponse
o
sinusoidal
a i-
a ion
o
inpu
signal
in
a
e men a ion
sys em
[19,
37,
38].
4.
Reassessmen
o
Con en ional
Models
o
P ocess
Con ol
A
la ge
numbe
o
models
ha e
been
p esen ed
as
su eyed
in
p e ious
i ems,
bu
i
is a
big
p oblem
o
choose
which
model
should
be
employed
o
"p ocess
con ol".
C i e ia
o
choice
o
model
may
be
lis ed
as
ollows:
1.
Objec
o
con ol:
wha
kind
o
con ol
is
desi ed;
holding
con ol
o
s a e
a iable
cons an ,
p og am
con ol
o
op imal
adap i e
con ol?
Mo e
sophis ica ed
models
mus
be
p epa ed
o
mo e
ine
con ol.
2.
A ailabili y
o
equipmen
o
acili ies
o
con ol:
a ailable
model
may
be
limi ed
wi h
ex en
o
ins umen a ion,
al hough
e e y
e o
should
be
made
o
sea ch
o
a
de ec o
o
con olle .
3.
A ailabili y
o
an
algo i hm
o
con ol:
a
con ol
algo i hm
is
ob iously
essen ial
o
ac ual
compu e
con ol,
he e o e
a
complex
model
can
no
be
employed.
Howe e ,
i is
no
so
di icul
o
de elop
a
con ol
algo i hm
i
he e
is
adequa e
echnical
alen .
Any
models
de eloped
p e i-
ously
o
o he
pu poses,
(e.g.,
kine ic
s udy,
undamen al
analysis
o
mic obial
beha iou
o
design
o
plan )
migh
be
analysed
and
modi ied
wi h
a
iew
o
applica ion
o
p ocess
con ol.
(1)
Applica ion
o
sensi i i y
analysis
o
assess
models
The
echnique
o
sensi i i y
analysis
has
been
applied
o
s udy
he
e ec
o
pa ame e
pe u ba ion
on
s a e
a iables
and
objec i e
unc ions
o
op imal
con ol.
Yoshida
and
Wada
[39]
analysed
he
beha iou
o
a
con inuous
cul u e
o
implemen a ion
o
p ocess
con ol
by
means
o
a
sensi i i y
analysis.
In
he
con inuous
e men a ion
sys em
whe e
he
pe o mance equa ion
is
exp essed
as
Eqs.
(7)
and
(8),
he
ollowing
equa ions
can
be
ob ained
o he
small
pe u ba ions,
x1
and
xz
o he
wo
s a e
a iables
X
and
S:
dxı
_
42
=
Yabxz
(22)
L
—