One App oach o Adap i e Con ol o a Tubula Chemical Reac o
PETR DOSTÁL, VLADIMÍR BOBÁL, JIŘÍ VOJTĚŠEK, and ZDENĚK BABÍK
Tomas Ba a Uni e si y in Zlin
Depa men o P ocess Con ol, Cen e o Polyme Sys ems
nam. T.G. Masa yka 5555, 760 01 Zlin
CZECH REPUBLIC
{dos alp, bobal, oj esek, babik}@ ai.u b.cz h p://www. ai.u b.cz/
Abs ac : – The pape deals wi h con inuous- ime adap i e con ol o a ubula chemical eac o wi h he
coun e cu en cooling as a nonlinea single inpu – single ou pu p ocess. The mean eac an empe a u e and
he ou pu eac an empe a u e a e chosen as he con olled ou pu s, and, he coolan low a e as he con ol
inpu . The nonlinea model o he eac o is app oxima ed by an ex e nal linea model wi h a s uc u e chosen
on he basis o con olled ou pu s s ep esponses. I s pa ame e s a e es ima ed ia co esponding del a model.
The con ol sys em s uc u e wi h wo eedback con olle s is conside ed. The esul ing con olle s a e de i ed
using polynomial app oach. The me hod is es ed on a ma hema ical model o he ubula chemical eac o .
Key-Wo ds: – Nonlinea sys em, ubula chemical eac o , app oxima e linea model, pa ame e iden i ica ion,
polynomial app oach, pole assignmen .
1 In oduc ion
Tubula chemical eac o a e uni s equen ly used
in chemical indus y. F om he sys em heo y poin
o iew, ubula chemical eac o s belong o a class
o nonlinea dis ibu ed pa ame e sys ems wi h
ma hema ical models desc ibed by se s o nonlinea
pa ial di e en ial equa ions (NPDRs). The me hods
o modelling and simula ion o such p ocesses a e
desc ibed e.g. in [1] – [5].
I is well known ha he con ol o chemical
eac o s, and, ubula eac o s especially, o en
ep esen s e y complex p oblem. The con ol
p oblems a e due o he p ocess nonlinea i y, i s
dis ibu ed na u e, and high sensi i i y o he s a e
and ou pu a iables o inpu changes. E iden ly, he
p ocess wi h such p ope ies is ha dly con ollable
by con en ional con ol me hods, and, i s e ec i e
con ol equi es applica ion some o ad anced
me hods. He e, a ious e icien me hods may be
used as he p edic i e con ol, e.g. [6], [7], [8], he
obus con ol, e.g. [9], he uzzy nonlinea con ol,
e.g. [10], he model e e ence con ol, e.g. [11], o
nonlinea con ol, e.g. [12], [13] and [14]. Some
o he s me hods a e desc ibed in [15].
One possible me hod o cope wi h his p oblem is
using adap i e s a egies based on an app op ia e
choice o a con inuous- ime ex e nal linea model
(CT ELM) wi h ecu si ely es ima ed pa ame e s.
These pa ame e s a e consequen ly used o pa allel
upda ing o con olle ‘s pa ame e s. Some esul s
ob ained in his ield we e p esen ed by au ho s o
his pape e.g. in [16] and [17].
Fo he CT ELM pa ame e es ima ion, ei he he
di ec me hod [18] and [19] o applica ion o an
ex e nal del a model wi h he same s uc u e as he
CT model can be used. The basics o del a models
ha e been desc ibed in e.g. [20] and [21]. Al hough
del a models belong in o disc e e models, hey do
no ha e such disad an ageous p ope ies connec ed
wi h sho ening o a sampling pe iod as disc e e z-
models. In addi ion, pa ame e s o del a models can
di ec ly be es ima ed om sampled signals.
Mo eo e , i can be easily p o ed ha hese
pa ame e s con e ge o pa ame e s o CT models o
a su icien ly small sampling pe iod (compa ed o
he dynamics o he con olled p ocess), as shown in
[22].
This pape deals wi h con inuous- ime adap i e
con ol o a ubula chemical eac o wi h a
coun e cu en cooling as a nonlinea single inpu –
single ou pu p ocess. Wi h espec o p ac ical
possibili ies o a measu emen and con ol, he mean
eac an empe a u e and he ou pu eac an
empe a u e a e chosen as he con olled ou pu s,
and, he coolan low a e as he con ol inpu . The
nonlinea model o he eac o is app oxima ed by a
CT ex e nal linea model wi h a s uc u e chosen on
he basis o compu ed con olled ou pu s s ep
esponses. The pa ame e s o he CT ELM hen a e
es ima ed ia co esponding del a model. The
con ol s uc u e wi h wo eedback con olle s is
conside ed, e.g. [23]. The esul ing con olle s a e
de i ed using he polynomial app oach [24] and he
pole assignmen me hod (see, e.g. [25]). The me hod
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is es ed on a ma hema ical model o a ubula
chemical eac o .
2 Model o he Reac o
An ideal plug- low ubula chemical eac o wi h a
simple exo he mic consecu i e eac ion 12
kk
ABC→→
in he liquid phase and wi h he coun e cu en
cooling is conside ed. Hea losses and hea
conduc ion along he me al walls o ubes a e
assumed o be negligible, bu dynamics o he me al
walls o ubes a e signi ican . All densi ies, hea
capaci ies, and hea ans e coe icien s a e
assumed o be cons an . Unde abo e assump ions,
he eac o model can be desc ibed by i e PDRs in
he o m
1
AA
A
cc
kc
z
∂∂
+=−
∂∂ (1)
12
BB
AB
cc
kckc
z
∂∂
+=−
∂∂ (2)
1
1
4()
() ()
w
p p
TTQ U
TT
zcdc
ρρ
∂∂
+= − −
∂∂ (3)
[
]
11
22
21
22
4()
()()
(
w w
pw
cw
TdU T T
dd c
dU T T
ρ
∂=−+
∂−
+−
(4)
12 2
22
312
4()
()()
cc
cwc
pc
TT ndU
TT
z
dnd c
ρ
∂∂
−= −
∂∂ − (5)
wi h ini ial condi ions
(,0) ()
s
AA
cz cz=, ( ,0) ( )
s
BB
cz cz=, ( ,0) ( )
s
Tz T z=,
(,0) ()
s
ww
Tz Tz=, ( ,0) ( )
s
cc
Tz T z=
and bounda y condi ions
0
(0, ) ( )
AA
c c =(kmol/m3),
0
(0, ) ( )
BB
c c =(kmol/m3), 0
(0, ) ( )
T T =(K),
(,) ()
ccL
TL T =(K).
He e, is he ime, z is he axial space a iable, c a e
concen a ions, T a e empe a u es, a e luid
eloci ies, d a e diame e s, ρ a e densi ies, cp a e
speci ic hea capaci ies, U a e hea ans e
coe icien s, n1 is he numbe o ubes and L is he
leng h o ubes. The subsc ip (⋅) s ands o he
eac an mix u e, (⋅)w o he me al walls o ubes,
(⋅)c o he coolan , and he supe sc ip (⋅)s o
s eady-s a e alues.
The eac ion a es and hea o eac ions a e
nonlinea unc ions exp essed as
0exp j
jj
E
kk
R
T
−
⎛⎞
=⎜⎟
⎝⎠
, j = 1, 2 (6)
11 2 2
()()
A B
QHkcHkc=−Δ +−Δ (7)
whe e k0 a e p e-exponen ial ac o s, E a e
ac i a ion ene gies, ()
H
−Δ a e in he nega i e
conside ed eac ion en alpies, and R is he gas
cons an .
The luid eloci ies a e calcula ed ia he eac an
and coolan low a es as
2
11
4
q
nd
π
= , 22
312
4
()
c
c
q
dnd
π
=− (8)
The pa ame e alues wi h co esponden uni s used
o simula ions a e gi en in Table 1.
Table 1. Used pa ame e alues
L = 8 m n1 = 1200
d1 = 0.02 m d2 = 0.024 m
d3 = 1 m
ρ = 985 kg/m3 cp = 4.05 kJ/kg K
ρw = 7800 kg/m3 cpw = 0.71 kJ/kg K
ρc = 998 kg/m3 cpc = 4.18 kJ/kg K
U1 = 2.8 kJ/m2s K U2 = 2.56 kJ/m2s K
k10 = 5.61⋅1016 1/s k20 = 1.128⋅1018 1/s
E1/R = 13477 K E2/R = 15290 K
(-ΔH 1) = 5.8⋅104 kJ/kmol (-ΔH 2) = 1.8⋅104 kJ/kmol
F om he sys em enginee ing poin o iew,
ou
(,)
AA
cL c=, ou
(,)
BB
cL c=, ou
(,)
TL T=
and ou
(0, )
cc
T T= a e he ou pu a iables, and,
()
q , ( )
c
q , 0()
A
c , 0()
T and ( )
cL
T a e he
inpu a iables. Among hem, o he con ol
pu poses, mos ly he coolan low a e can be aken
in o accoun as he con ol a iable, whe eas o he
inpu s en e ing in o he p ocess can be accep ed as
dis u bances. In his pape , he mean eac an
empe a u e gi en by
0
1
() (,)
L
m
T Tz dz
L
=∫ (9)
and he eac an ou pu empe a u e ou ()
T a e
conside ed as he con olled ou pu s.
3 Compu a ion Models
Fo compu a ion o bo h s eady-s a e and dynamic
cha ac e is ics, he ini e di e ences me hod is
employed. The p ocedu e is based on subs i u ion o
he space in e al 0,zL∈< > by a se o disc e e
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node poin s
{
}
i
z o i = 1, … , n , and, subsequen ly,
by app oxima ion o de i a i es wi h espec o he
space a iable in each node poin by ini e
di e ences. Two ypes o ini e di e ences a e
applied, ei he he backwa d ini e di e ence
1
(,) ( ,)
(,)
(,) ( 1,)
i
ii
zz
yz yz
yz
zh
yi yi
h
−
=
−
∂≈=
∂
−−
=
(10)
o he o wa d ini e di e ence
1
(,)(,)
(,)
(1,) (,)
i
ii
zz
yz yz
yz
zh
yi yi
h
+
=
−
∂≈=
∂
+−
=
. (11)
He e, a unc ion ( , )yz is con inuously
di e en iable in he in e al 0,L<>
, and, hLn=
is he disk e iza ion s ep.
3.1 Dynamic model
Applying he subs i u ions (10), (11) in (1) – (5)
and, omi ing he a gumen in pa en hesis, PDRs
(1) – (5) a e app oxima ed by a se o ODRs in he
o m
[]
01 0
() () () ( 1)
AAA
dc i bkicibci
d =− + + − (12)
[]
102
0
() () () () ()
(1)
BAB
B
dc i kic i b kic i
d
bc i
=−+ +
+−
(13)
1020
2
() () ( ) () ( 1)
()
w
dT i bQ i b b T i b T i
d
bT i
=−+ +−+
+
(14)
[][]
34
() () () () ()
w w cw
dT i bTi Ti bTi T i
d =−+− (15)
56 5
6
() ()()(1)
()
ccc
w
dT m bbTmbTm
d
bT m
=− + + + +
+
(16)
o 1,...,in= and 1mni=−+, and, wi h ini ial
condi ions
(,0) ()
s
AA
ci ci=, (,0) ()
s
BB
ci ci=, (,0) ()
s
Ti T i=,
(,0) ()
s
ww
Ti Ti= and ( ,0) ( )
s
cc
Ti T i= o 1,...,in=.
The bounda y condi ions en e in o Eqs. (12) – (14)
and (16) o i = 1 .
Now, nonlinea unc ions in Eqs. (12) – (16) ake
he disc e e o m
0
() exp ()
j
jj
E
ki k
R
Ti
−
⎛⎞
=⎜⎟
⎝⎠
, j = 1, 2 (17)
11
22
() ( ) () ()
( ) () ()
A
B
Qi H kic i
Hkici
=−Δ +
+−Δ (18)
o i = 1, … , n.
The pa ame e s b in Eqs. (12) – (16) a e calcula ed
om o mulas
0
bh
=, 1
1
()
p
bc
ρ
=, 1
21
4
()
p
U
bdc
ρ
=,
11
322
21
4
()()
p
w
dU
bdd c
ρ
=−, 22
422
21
4
()()
p
w
dU
bdd c
ρ
=− (19)
5c
bh
=, 12 2
622
312
4
()()
p
c
ndU
bdnd c
ρ
=−.
He e, he o mulas o compu a ion o Tm and ou
T
ake he disc e e o m
1
1
() ( ,)
n
m i
i
T Tz
n=
=∑, ou () ( ,)
n
T Tz = (20)
3.2 S eady-s a e model
Compu a ion o he s eady-s a e cha ac e is ics is
necessa y no only o a s eady-s a e analysis bu he
s eady s a e alues ( )
s
yi also cons i u e ini ial
condi ions in ODRs (12) – (16) (he e, y p esen s
some o he a iable in he se (12) – (16)).
The s eady-s a e model can simply be de i ed
equa ing he ime de i a i es in (12) – (16) o ze o.
Then, a e some algeb aic manipula ions, he
s eady-s a e model akes he o m o di e ence
equa ions
0
01
() ( 1)
()
ss
AA
s
b
ci ci
bki
=−
+ (21)
10
02
1
() () () ( 1)
()
ssss
BAB
s
ci kici bci
bki
⎡
⎤
=+−
⎣
⎦
+ (22)
10 2
02
1
() () ( 1) ()
ssss
w
Ti bQi bTi bTi
bb
⎡
⎤
=+−+
⎣
⎦
+ (23)
34
34
1
() () ()
sss
w c
Ti bTi bTi
bb
⎡
⎤
=+
⎣
⎦
+ (24)
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56
56
1
() ( 1) ()
sss
ccw
Tm bTm bTm
bb
⎡⎤
=++
⎣⎦
+ (25)
o 1,...,in= and 1mni=−+. Nonlinea unc ions
acco dan wi h a s eady-s a e a e
0
() exp ()
j
s
jj s
E
ki k
R
Ti
⎛⎞
−
=⎜⎟
⎜⎟
⎝⎠
, j = 1, 2 (26)
11
22
() ( ) () ()
( ) () ()
sss
A
ss
B
Qi H kici
Hkici
=−Δ +
+−Δ
(27)
Now, he o mulas o compu a ion T
m and ou
T
ha e he o m
1
1()
n
ss
m i
i
TTz
n=
=∑, ou ()
ss
n
TTz= (28)
3.3 S eady-s a e and dynamic cha ac e is ics
Typical eac an empe a u e p o iles along he
eac o ubes compu ed o 02.85
s
A
c=, 00
s
B
c=,
0323
s
T=, 0293
s
c
T= and 0.15
s
q= o a ious
coolan low a es a e shown in Fig. 1. A p esence
o a maximum on he eac an empe a u e p o iles
is a common p ope y o many ubula eac o s wi h
exo he mic eac ions.
012345678
320
330
340
350
360
Reac an empe a u e (K)
z (m)
1 - qs
c = 0.2
2 - qs
c = 0.25
3 - qs
c = 0.3
1
2
3
Fig. 1 Reac an empe a u e p o iles o a ious
coolan low a es.
A dependences o he eac an mean empe a u e
and he eac an ou pu empe a u e on he coolan
low a e is shown in Fig. 2. The o m o bo h
cu es documen s a nonlinea ela ion be ween
supposed con olled ou pu s and he coolan low
a e which is conside ed as he con ol inpu .
Dynamic cha ak e is ics we e compu ed in he
neighbou hood o he chosen ope a ing poin
3
0.27 m /s
s
c
q=, 334.44K
s
m
T=, ou 326.10K
s
T=
Fo he dynamic analysis and subsequen con ol
pu poses, he con olled ou pu s a e de ined as
0.20 0.25 0.30 0.35 0.40
310
315
320
325
330
335
340
345
Ts
m
Ts
ou
Ts
ou = 326.10
Tempe a u es (K)
qs
c (m3/s)
qs
c = 0.27
Ts
m = 334.44
Fig. 2 Dependence o he eac an mean
empe a u e on he coolan low a es.
de ia ions om s eady alues
1
2 ou ou ou
() () ()
() () ()
s
mmm
s
y T T T
y T T T
=Δ = −
=Δ = − . (29)
Such o m is equen ly used in he con ol. The
de ia ion o he coolan low a e is deno ed as
()
s
cc c
qq qΔ= −. (30)
The esponses o bo h ou pu s o he coolan low
a e s ep changes a e shown in Figs. 3, 4.
0 50 100 150 200
-12
-10
-8
-6
-4
-2
0
2
4
6
8
y1 (K)
(s)
1 - Δqc = - 0.05
2 - Δqc = - 0.025
3 - Δqc = 0.025
4 - Δqc = 0.05
1
2
3
4
Fig. 3 Reac an mean empe a u e s ep esponses.
0 50 100 150 200
-10
-8
-6
-4
-2
0
2
4
y2 (K)
(s)
1 - Δqc = - 0.05
2 - Δqc = - 0.025
3 - Δqc = 0.025
4 - Δqc = 0.05
1
2
3
4
Fig. 4 Reac an ou pu empe a u e s ep esponses.
The abo e shown esponses demons a e mo e
exp essi e nonlinea beha iou o he eac an
ou pu empe a u e o inpu changes han he
eac an mean empe a u e. This ac is e iden also
om he gain alues compu ed as
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()
lim
s c
y
gq
→∞
=Δ (31)
and p esen ed in Tab. 2.
Tab. 2 Gains o a ious inpu hanges.
Δqc - 0.025 - 0.05 0.025 0.05
Main eac an empe a u e
gs -155.4 -166.2 -263.5 -205.1
Ou pu eac an empe a u e
gs -69.6 -72.0 -194.3 -193.5
This ac p edica es be e p ope ies o he eac an
mean empe a u e as he con olled ou pu han he
eac an ou pu empe a u e. Mo eo e , he
dynamics o he eac an ou pu empe a u e is
slowe in compa ison wi h he dynamics o he
eac an mean empe a u e.
4 CT and Del a ELM
Fo he con ol pu poses, he con ol inpu a iable
a e conside ed in he o m
()
() 10
s
cc
s
c
q q
u
q
−
= (32)
This exp ession enables o ob ain con ol inpu and
con olled ou pu a iables o app oxima ely he
same magni ude.
A choice o he CT ELM s uc u e does no s em
om known s uc u e o he model (1) – (5) bu
om a cha ac e o simula ed s ep esponses. I is
well known ha in adap i e con ol a con olled
p ocess o a highe o de can be app oxima ed by a
linea model o a lowe o de wi h a iable
pa ame e s. Taking in o accoun p o iles o cu es
in Figs. 3 and 4 wi h ze o de i a i es in = 0, he
second o de CT ELM has been chosen o bo h
con olled ou pu s in he o m o he second o de
linea di e en ial equa ion
100
() () () ()y ay a y bu ++ =
  (33)
whe e y = y1 o y = y2 , and, in he complex domain,
as he ans e unc ion
0
210
() b
Gs
s
as a
=++
. (34)
Es ablishing he δ ope a o
0
1q
T
δ
−
= (35)
whe e q is he o wa d shi ope a o and T0 is he
sampling pe iod, he del a ELM co esponding o
(33) akes he o m
2100
() () () ()y a y a y bu
δδ
′′ ′′′′′
++=
(36)
whe e ′is he disc e e ime.
When he sampling pe iod is sho ened, he del a
ope a o app oaches he de i a i e ope a o , and, he
es ima ed pa ame e s ,ab
′′
each he pa ame e s a,
b o he CT model (33).
5 Del a Model Pa ame e Es ima ion
Subs i u ing 2 k
′=−, equa ion (36) can be
ew i en o he o m
210
0
(2) (2) (2)
(2)
yk a yk a yk
buk
δδ
′′
−+ −+ −=
′
=−
(37)
In he pape , he ecu si e iden i ica ion me hod
wi h exponen ial and di ec ional o ge ing was
used.
Es ablishing he eg ession ec o
()
(1) (2) (2)(2)
Tkykykuk
δ
δ
−=− − − − −Φ(38)
whe e
0
(1)(2)
(2)
yk yk
yk T
δ
−− −
−= (39)
he ec o o del a model pa ame e s
()
100
()
Tkaab
δ
′′′
=Θ (40)
is ecu si ely es ima ed om he ARX model
2(2) ()(1)()
T
yk k k k
δδ
δε
−= −+ΘΦ (41)
whe e
2
2
0
() 2( 1) ( 2)
(2)
yk yk yk
yk T
δ
−−+−
−= (42)
6 Con olle Design
The con ol sys em wi h wo eedback con olle s is
depic ed in Fig. 5.
-
-
e
wu
u
0
y
R
CT ELM
Q
Fig. 5. Con ol sys em wi h wo eedback con olle s
In he scheme, w is he e e ence signal, deno es
he load dis u bance, e he acking e o , u0 ou pu
o con olle s, u he con ol inpu and y he
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con olled ou pu . The ans e unc ion G(s) o
he CT ELM is gi en by (34).
The e e ence w and he dis u bance a e
conside ed as s ep unc ions wi h ans o ms
0
() w
Ws
s
=, 0
()
Vs
s
= (43)
The ans e unc ions o bo h con olle s a e in
o ms
() ()
() , ()
() ()
s qs
Rs Qs
p
sps
==


(44)
whe e q
, and
p
 a e cop ime polynomials in s
ul illing he condi ion o p ope ness deg deg
p≤
and deg deg
qp≤ .
The con olle design desc ibed in his sec ion
appea s om he polynomial app oach. The gene al
equi emen s on he con ol sys em a e o mula ed
as i s in e nal p ope ness and s ong s abili y (in
addi ion o he con ol sys em s abili y, also he
con olle s abili y is equi ed), asymp o ic acking
o he e e ence and load dis u bance a enua ion.
The p ocedu e o de i e admissible con olle s can
be pe o med as ollows:
T ans o ms o he basic signals in he closed-loop
sys em ake ollowing o ms ( o simpli ica ion, he
a gumen s is in some equa ions omi ed)
[]
() () ()
b
Ys Ws pVs
d
=+
 (45)
[]
1
() ( ) () ()
E
sapbqWsbpVs
d
=+ −
  (46)
[]
() () ()
a
Us Ws pVs
d
=+
 (47)
He e,
[
]
() () () () () ()ds as ps bs s qs=++

(48)
is he cha ac e is ic polynomial wi h oo s as poles
o he closed-loop.
Es ablishing he polynomial as
() () ()
s s qs=+
 (49)
and subs i u ing (49) in o (48), he condi ion o he
con ol sys em s abili y is ensu ed when
polynomials
p
 and a e gi en by a solu ion o he
polynomial Diophan ine equa ion
() () ()() ()
asps bs s ds+=
 (50)
wi h a s able polynomial d on he igh side.
Wi h ega d o he ans o ms (43), he asymp o ic
acking and load dis u bance a enua ion a e
p o ided by di isibili y o bo h e ms ap bq+

and
p
 in (46) by s. This condi ion is ul illed when
polynomials
p
and q
ha e o ms
() ()
p
ssps=
, ( ) ( )
qs sqs=
. (51)
Subsequen ly, he ans e unc ions (44) ake o ms
()
() ()
qs
Qs
p
s
=, ()
() ()
s
Rs
s
ps
= (52)
and, a s able polynomial p(s) in hei denomina o s
ensu es he s abili y o con olle s.
The con ol sys em sa is ies he condi ion o in e nal
p ope ness when he ans e unc ions o all i s
componen s a e p ope . Consequen ly, he deg ees
o polynomials q and mus ul il hese inequali ies
deg deg
qp≤, deg deg 1 p≤+. (53)
Now, he polynomial can be ew i en o he o m
() () ()
s s sqs=+ . (54)
Taking in o accoun he sol abili y o (50) and
condi ions (53), he deg ees o polynomials in (50)
and (52) can be easily de i ed as
deg deg deg
a==, deg deg 1qa=−,
deg deg 1
p
a≥−, deg 2degda≥. (55)
Deno ing deg a = n, polynomials , and q ha e
o ms
0
()
n
i
i
i
s s
=
=
∑
,
0
()
n
i
i
i
s s
=
=
∑
, 1
1
()
n
i
i
i
qs qs−
=
=∑ (56)
and, ela ions among hei coe icien s a e
00
=, iii
q +=
o 1,...,in=. (57)
Since by a solu ion o he polynomial equa ion (50)
p o ides calcula ion o coe icien s i, unknown
coe icien s i and qi can be ob ained by a choice o
selec able coe icien s 0,1
i
β
∈ such ha
iii
β
=, (1 )
iii
q
β
=− o 1,...,in=. (58)
The coe icien s i
β
dis ibu e a weigh be ween
nume a o s o ans e unc ions Q and R.
Rema k: I 1
i
β
= o all i, he con ol sys em in Fig.
5 educes o he 1DOF con ol con igu a ion (Q =
0). I 0
i
β
= o all i, and, bo h e e ence and load
dis u bance a e s ep unc ions, he con ol sys em
co esponds o he 2DOF con ol con igu a ion.
Fo he second o de model (34) wi h deg 2
a=, he
con olle 's ans e unc ions ake speci ic o ms
21
0
2
210
0
()
() ()
()
() () ( )
qs qs q
Qs ps s p
s s
s
Rs sps s s p
+
==
+
++
==
+
. (59)
whe e
00
=, 111
β
=, 222
β
=,
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111
(1 )q
β
=− , 222
(1 )q
β
=− . (60)
The con olle pa ame e s hen esul om a
solu ion o he polynomial equa ion (50) and depend
upon coe icien s o he polynomial d. The nex
p oblem he e is o ind a s able polynomial d ha
enables o ob ain accep able s abilizing con olle s.
In his pape , he polynomial d wi h oo s
de e mining he closed-loop poles is chosen as
2
() ()( )ds ns s
α
=+ (61)
whe e n is a s able polynomial ob ained by spec al
ac o iza ion
() () () ()asas nsns
∗∗
= (62)
and α is he selec able pa ame e .
No e ha a choice o d in he o m (61) p o ides he
con ol o a good quali y o ape iodic con olled
p ocesses.
The coe icien s o n hen a e exp essed as
2
00
na=, 2
1100
22nana=+−
(63)
and, he con olle pa ame e s p0 and can be
ob ained om solu ion o he ma ix equa ion
10
00
0
1000
00
00
000
ab
ab
b
⎛⎞
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎝⎠
×
0
2
1
0
p
⎛⎞
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎝⎠
=
31
20
1
0
da
da
d
d
−
⎛⎞
⎜⎟
−
⎜⎟
⎜⎟
⎜⎟
⎝⎠
(64)
whe e
2
31 2 10
22
10100
2, 2
2,
dn d nn
dnndn
αα α
αα α
=+ = ++
=+ =
(65)
Now, i ollows om he abo e in oduced
p ocedu e ha uning o con olle s can be
pe o med by a sui able choice o selec able
pa ame e s β and α.
The con olle pa ame e s and q can hen be
ob ained om (60).
The adap i e con ol sys em is shown in Fig. 6.
Con olle
Pa ame e es ima ion
Con olled
p ocess
Compu a ion o
con olle pa ame e s
T
0
T
0
w
u
q, p
y
b, a
-
Fig. 6 Adap i e con ol scheme.
7 Con ol Simula ion
Also he con ol simula ions we e pe o med in a
neighbou hood o he ope a ing poin
3
0.27 m /s
s
c
q=, 334.44K
s
m
T=, ou 326.10K
s
T=.
Fo he s a ( he adap a ion phase), he P con olle
wi h a small gain was used in all simula ions.
Wi h espec o mo e exp essi e nonlinea i y and
slowe dynamics o he eac an ou pu empe a u e
in compa ison wi h he eac an mean empe a u e,
he changes o e e ences as well as he con ol
unning ime in e als we e chosen di e en o
bo h ou pu s.
The e ec o he pole α on he con olled esponses
is anspa en om Figs. 7 and 8. Fo bo h ou pu s,
wo alues o α we e selec ed. The con ol
simula ions show sensi i i y o con olled ou pu s o
α. The highe alues o his pa ame e speed he
con ol, howe e , hey p o ide g ea e o e shoo s
(unde shoo s). O he he e no men io ed simula ions
showed ha a ca eless selec ion o he pa ame e α
can lead o con olled ou pu esponses o a poo
quali y, o oscilla ions o e en o he con ol
ins abili y.
0 100 200 300 400 500 600 700 800
-8
-6
-4
-2
0
2
4
6
y1 (K)
(s)
1 - α = 0.075
2 - α = 0.2
1
2
1
2
w
Fig. 7 Con olled ou pu y1 esponses: e ec o α
(β1 = 1, β2 = 0.5).
0 200 400 600 800 1000 1200
-2
-1
0
1
2
y2 (K)
(s)
1 - α = 0.05
2 - α = 0.025
w
1
2
1
21
2
Fig. 8 Con olled ou pu y2 esponses: e ec o α
(β1 = 1, β2 = 0).
Mo eo e , an inc easing α leads o highe alues
and changes o he con ol inpu as shown in Fig. 9
and 10. This ac can be impo an in con ol o eal
echnological p ocesses.
The con olled ou pu y1 esponse o wo alues β2
is shown in Fig. 11. I can be seen ha an e ec o
his pa ame e e is insigni ican .
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0 100 200 300 400 500 600 700 800
0.21
0.22
0.23
0.24
0.25
0.26
0.27
0.28
0.29
0.30
0.31
0.32
0.33
qc (m3/ s)
(s)
1 - α = 0.075
2 - α = 0.2
1
1
1
2
2
2
Fig. 9 Coolan low a e esponses in con ol o
eac an mean empe a u e – e ec o α
(β1 = 1, β2 = 0.5).
0 200 400 600 800 1000 1200
0.22
0.24
0.26
0.28
0.30
qc (m3/ s)
(s)
1 - α = 0.05
2 - α = 0.025
1
2
1
2
1
2
Fig. 10 Coolan low a e esponses in con ol o
eac an ou pu empe a u e – e ec o α
(β1 = 1, β2 = 0).
0 100 200 300 400 500 600 700 800
-6
-4
-2
0
2
4
y1 (K)
(s)
1 - β2 = 0
2 - β2 = 1
w
1
2
Fig. 11 Con olled ou pu esponses: e ec o β2
(α = 0.1, β1 = 1).
The con olled ou pu esponses documen ing an
e ec o he pa ame e β1 a e in Figs. 12 and 13. In
bo h cases, a highe alue o β1 esul s in g ea e
o e shoo s (unde shoo s) whe eas i s in luence on
he speed o con ol is inexp essi e.
Co esponding con ol inpu esponses can be seen
in Figs. 14 and 15. The e, an inc easing β1 leads o
g ea e alues o inpu s, howe e , i can educe
occu ed oscila ions, as shown in Fig. 15.
O in e es , he e olu ion o es ima ed CT ELM
pa ame e s in con ol o he eac an mean
empe a u e is shown in Fig. 16.
0 100 200 300 400 500 600 700 800
-8
-6
-4
-2
0
2
4
6
y1 (K)
(s)
1 - β1 = 0.2
2 - β1 = 1
w
1
2
1
2
Fig. 12 Con olled ou pu esponses: e ec o β1
(α = 0.15, β2 = 0).
0 200 400 600 800 1000 1200 1400
-2
-1
0
1
2
y2 (K)
(s)
1 - β1 = 0
2 - β1 = 1
w
1
1
1
2
2
2
Fig. 13 Con olled ou pu esponses: e ec o β1
(α = 0.04, β2 = 0).
0 100 200 300 400 500 600 700 800
0.22
0.23
0.24
0.25
0.26
0.27
0.28
0.29
0.30
0.31
0.32
qc (m3/ s)
(s)
1 - β1 = 0.2
2 - β1 = 1
1
1
1
2
2
2
Fig. 14 Coolan low a e esponses in con ol o
eac an mean empe a u e – e ec o β1
(α = 0.15, β2 = 0).
0 200 400 600 800 1000 1200 1400
0.23
0.24
0.25
0.26
0.27
0.28
0.29
0.30
qc (m3/ s)
(s)
1 - β1 = 0
2 - β1 = 1
1
11
2
2
2
Fig. 15 Coolan low a e esponses in con ol o
eac an ou pu empe a u e – e ec o β1
(α = 0.15, β2 = 0).
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0 100 200 300 400 500 600 700 800
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
CT ELM pa ame e s
(s)
1 - b0
2 - 10 x a0
3 - 10-1x a1
1
2
3
Fig. 16 CT ELM pa ame e e olu ion (α = 0.15,
β1 = 1, β2 = 0).
A p esence o an in eg a ing pa in he con olle
enables ejec ion o a ious s ep dis u bances
en e ing in o he p ocess. As an example, s ep
dis u bances a enua ion o he ou pu y1 is
p esen ed. S ep dis u bances 3
00.15 kmol/ m
A
cΔ= ,
3
0.03 m /s
qΔ=− and 02K
TΔ= we e injec ed
in o he nonlinea model o he eac o in imes
220s
=, 440s
= and 640s
=. The con olle
pa ame e s we e es ima ed only in he i s
( acking) in e al < 200 s. The au ho s'
expe iences p o ed ha an u iliza ion o ecu si e
iden i ica ion using he del a model a e eaching o
a cons an e e ence and in p esence o s ep
dis u bances dec eases he con ol quali y. F om his
eason, du ing in e al ≥ 200 s, ixed pa ame e s
we e used. The con olled ou pu esponses y1 a e
shown in Fig. 17.
0 100 200 300 400 500 600 700 800 900
0
1
2
3
4
5
6
7
y1 (K)
(s)
w
Fig. 17 Con olled ou pu in p esence o s ep
dis u bances (α = 0.15, β1 = 0.5, β2 = 0).
To illus a e an e ec o an addi i e andom
dis u bance, he esul o he con olled ou pu y1
simula ion in a p esence o he andom signal
0
() ()
s
A
A
c c=− is shown in Fig. 18.
8 Conclusions
In his pape , one app oach o con inuous- ime
adap i e con ol o he mean and ou pu eac an
-0.04
-0.02
0.00
0.02
0.04
(kmol/ m3)
0 100 200 300 400 500 600 700 800
-2
-1
0
1
2
3
4
y1 (K)
(s)
w
Fig. 18. Con olled ou pu in he p esence o
andom dis u bance in 0
A
c (α = 0.15).
empe a u es in a ubula chemical eac o was
p oposed. The con ol s a egy is based on he
p elimina y s eady-s a e and dynamic analysis o he
p ocess and on he assump ion o he empe a u e
measu emen along he eac o . The p oposed
algo i hm employs an al e na i e con inuous- ime
ex e nal linea model wi h pa ame e s ob ained
h ough ecu si e pa ame e es ima ion o a
co esponding del a model. The con ol sys em
s uc u e wi h wo eedback con olle s is
conside ed. Resul ing con inuous- ime con olle s
a e de i ed using he polynomial app oach and
gi en by a solu ion o he polynomial Diophan ine
equa ion. Tuning o hei pa ame e s is possible ia
closed-loop pole assignmen . The p esen ed me hod
has been es ed by compu e simula ion on he
nonlinea model o he ubula chemical eac o wi h
a consecu i e exo he mic eac ion. The simula ion
esul s demons a e an applicabili y o he p esen ed
con ol s a egy.
Acknowledgmen s
The au ho s wish o hank o he Minis y o
Educa ion o he Czech Republic
(MSM7088352101) o inancial suppo . This
a icle was c ea ed wi h suppo o Ope a ional
P og amme Resea ch and De elopmen o
Inno a ions co- unded by he Eu opean Regional
De elopmen Fund (ERDF) and na ional budge o
Czech Republic wi hin he amewo k o he Cen e
o Polyme Sys ems p ojec ( eg.numbe :
CZ.1.05/2.1.00/03.0111).
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