Min-Max P edic i e Con ol o a
Pilo Plan using a QP App oach
J.K. G ube , D.R. Rami ez, T. Alamo, C. Bo dons, E.F. Camacho
Abs ac —The p ac ical implemen a ion o Min-Max MPC
(MMMPC) con olle s is limi ed by he compu a ional bu den
equi ed o compu e he con ol law. This p oblem can be
ci cum en ed by using app oxima e solu ions o uppe bounds
o he wo s possible case o he pe o mance index. In a
p e ious wo k, he au ho s p esen ed a compu a ionally e icien
MMMPC con ol s a egy in which a close app oxima ion o
he solu ion o he min-max p oblem is compu ed using a
quad a ic p og amming p oblem. In his pape , his app oach
is alida ed h ough i s applica ion o a pilo plan in which
he empe a u e o a eac o is con olled. The pilo plan is
ope a ed wi h a Sima ic IT SCADA sys em. The con olle has
been implemen ed in Ma lab and connec ed o he SCADA
using OPC. Realis ic alues o he pa ame e s o he MMMPC
con olle ha e been used. The beha io o he sys em and he
con olle is illus a ed by means o expe imen al esul s.
I. INTRODUCTION
The idea behind min-max model p edic i e con olle s
(MMMPC) is no new ([4]). In hese con olle s, he con ol
signal is compu ed o he wo s case o a cos unc ion
ha conside s he e ec o p ocess model unce ain ies
and dis u bances in he con olle pe o mance. The main
d awback o his app oach is he compu a ional bu den ha
akes o compu e he con ol signal. This usually in ol es he
solu ion o an NP-ha d min-max p oblem ([10], [16]). As a
esul , he numbe o applica ions o hese con ol s a egies
is e y small, e en when he e is e idence ha hey wo k
be e han s anda d p edic i e con olle s in p ocesses wi h
unce ain dynamics [5], [8].
Mul i-pa ame ic p og amming has been applied o show
ha he MMMPC con ol law is piecewise a ine when a
quad a ic ([14]) o 1-no m based c i e ion ([2], [7]) is usedas
cos unc ion. Thus, explici o ms o he con ol law can be
buil . Such explici o ms can be e alua ed e y as p o ided
ha he complexi y o he s a e space pa i ion is mode a e,
which is he case o many applica ions. Howe e , i he
p ocess model o he con olle uning pa ame e s change,
he compu a ion o he con olle has o be edone.
Acommonsolu ion o hecompu a ionalbu denissueis
o use an uppe bound o he wo s case cos ins ead o com-
pu ing i explici ly. This uppe bound can be compu ed by
using linea ma ix inequali ies (LMI) echniques such as in
[9], [11]. Howe e , he LMI p oblems ha e a compu a ional
bu den ha canno be neglec ed in ce ain applica ions. In [1]
adi e en app oachbasedonacompu a ionallycheapuppe
bound o he wo s case cos is p esen ed. In ha wo k, he
The au ho s a e wi h he Dep . de Ingenie ´ıa de Sis emas y
Au om´a ica, Escuela Supe io de Ingenie os, Uni e si y o Se ille,Spain
{jg ube , dani , alamo}@ca uja.us.es, {bo dons,
edua do}@esi.us.es
min-max p oblem is eplaced by a quad a ic p og amming
(QP) p oblem ha p o ides a close app oxima ion o he
solu ion o he o iginal min-max p oblem. The compu a ional
bu den is much lowe han ha o he min-max p oblem and
is compa able o ha o a s anda d cons ained MPC based
on a quad a ic cos unc ion. Thus, i can be easily imple-
men ed in almos any pla o m capable o un a cons ained
MPC. Also, s abili y o he p oposed app oach is gua an eed.
In his wo k, he app oach p esen ed in [1] has been
alida ed by means o i s applica ion o a pilo plan . The
pilo plan is used o simula e an exo he mic chemical
eac ion wi h nonlinea dynamics. This p ocess has been used
in p e ious wo ks, hus he expe imen al esul s p esen ed
can be compa ed wi h o he s a egies such as nonlinea and
linea p edic i e con ol [6]. In he expe imen s, es ic ions
in he con ol ac ion and he ou pu ha e been conside ed.
The esul s ob ained p o e he alidi y o he con ol s a egy.
The low compu a ional bu den o he con ol s a egy applied
o he pilo plan allows ealis ic alues o he con ol
and p edic ion ho izons (i.e., he pa ame e s on which he
compu a ional bu den depends). I is also no ewo hy ha he
compu e on which he p edic i e con ol algo i hm is imple-
men ed and simul aneously execu es he SCADA Sima ic-IT,
does no ha e su icien calcula ion powe o implemen a
con en ional min-max p edic i e con ol s a egy. The e o e
acompu a ionallye icien s a egyas ha usedin hiswo k
is a good choice i he use o his ype o con ol is desi ed.
The pape is o ganized as ollows: sec ion II p esen s
he MMMPC s a egy. Sec ion III p esen s he p oposed
implemen a ion s a egy. In sec ion IV a de ailed desc ip ion
o he used pilo plan is gi en. The s a egy is illus a ed by
means o expe imen al esul s o he pilo plan in sec ion V.
Finally, sec ion VI p esen s some conclusions.
II. MIN-MAX MPC WITH BOUNDED ADDITIVE
UNCERTAINTIES
Conside he ollowing s a e space model wi h bounded
addi i e unce ain ies ([3]):
x( +1)=Ax( )+Bu( )+D
θ
( +1)(1)
wi h x( )∈Rdimx he s a e ec o , u( )∈Rdimu he inpu
ec o and
θ
( )∈{
θ
∈Rdim
θ
:∥
θ
∥∞≤
ε
} he unce ain y,
ha is supposed o be bounded. The sys em is subjec o p
s a e and inpu ime in a ian cons ain s Fuu( )+Fxx( )≤g
whe e Fu∈Rp×dimu and Fx∈Rp×dimx.I isassumedasemi-
eedback app oach in which he con ol inpu is gi en by
u( )=−Kx( )+ ( ),(2)
P ep in submi ed o 47 h IEEE Con e ence on Decision and Con ol.
Recei ed Ma ch 10, 2008.
whe e he eedback ma ix Kis chosen o achie e some
desi ed p ope y such as nominal s abili y o LQR op imali y
wi hou cons ain s. The MMMPC con olle will compu e
he op imal sequence o co ec ion con ol inpu s ( ).The
s a e equa ion o sys em (1) can be ew i en as
x( +1)=ACLx( )+B ( )+D
θ
( +1),ACL =(A−BK).(3)
The p oposed s a egy also wo ks wi hou semi- eedback
app oach (i.e., u( )= ( )). All he compu a ional ad an ages
o he s a egy emain he same. Fu he mo e i he p ocess
is open-loop s able (as in he case o pilo plan used in
his wo k) he s abilizing condi ions, which will be discussed
la e , can be used wi hou p oblems.
The cos unc ion is a quad a ic pe o mance index:
V(x, ,
θ
θ
θ
)=
N−1
∑
j=0
x( +j| )TQx( +j| )+ (4)
N−1
∑
j=0
u( +j| )TRu( +j| )
+x( +N| )TPx( +N| )
whe e x( | )=x,x( +j| )is he p edic ion o he s a e
o +jmade a and u( +j| )=−Kx( +j| )+ ( +
j| ).No e ha hese aluesdependon he u u e al-
ues o he unce ain y. The sequence o u u e alues o
θ
( )o e a p edic ion ho izon Nis deno ed by
θ
θ
θ
=
!
θ
( +1)T,···,
θ
( +N)T"T,andΘ
Θ
Θ={
θ
θ
θ
∈RN·dim
θ
:∥
θ
θ
θ
∥∞≤
ε
}is he se o possible unce ain y ajec o ies. On he
o he hand, =! ( | )T,···, ( +N−1| )T"Tis he con ol
co ec ion sequence. Ma ices Q,P∈Rdimx×dimx and R∈
Rdimu×dimu a e symme ic posi i e de ini e ma ices used as
weigh ing pa ame e s.
Min-Max MPC ([4]) minimizes he cos unc ion o he
wo s possible case o he p edic ed u u e e olu ion o he
p ocess s a e o ou pu signal. This is accomplished h ough
he solu ion o a min-max p oblem:
∗(x)= a gmin
max
θ
θ
θ
∈ΘV(x, ,
θ
θ
θ
)
s. .Fuu( +j| )+Fxx( +j| )≤g,
j=0,...,N,∀
θ
θ
θ
∈Θ,
x( +N| )∈Ω,∀
θ
θ
θ
∈Θ,
(5)
A e minal egioncons ain x( +N| )∈Ω,whe eΩis a
polyhed on, is included o assu e s abili y o he con ol law
([12]).
The p edic ions x( +j| )and u( +j| )depend linea ly on
x, and
θ
θ
θ
.Thismeans ha i ispossible o inda ec o
d∈Rpand ma ices Gx,G and G
θ
,such ha all he obus
linea cons ain s o p oblem (5) can be ew i en as:
Gi
xx+Gi
+Gi
θθ
θ
θ
≤di,i=1...,p,∀
θ
θ
θ
∈Θ,
whe e Gi
x,Gi
,Gi
θ
deno e he i- h ows o Gx,G and G
θ
espec i ely and diis he i- h componen o d∈Rp.Deno e
now ∥Gi
θ
∥1 he sum o he absolu e alues o ow Gi
θ
.
Taking in o accoun ha max
θ
θ
θ
∈ΘGi
θθ
θ
θ
=max∥
θ
θ
θ
∥∞≤
ε
Gi
θθ
θ
θ
=
ε
∥Gi
θ
∥1, he obus ul illmen o hecons ain sissa is ied
i and only i Gi
xx+Gi
+
ε
∥Gi
θ
∥1≤di,i=1,...,p.
The e o e, o gua an ee obus cons ain sa is ac ion, he
ollowing se o linea cons ain s mus be sa is ied:
Gxx+G ≤d
ε
.
whe e he i- h componen o d
ε
is equal o di−
ε
∥Gi
θ
∥1.No e
ha his is a necessa y and su icien condi ion.
Taking in o accoun (3),(2) and (4), he cos unc ion can
be e alua ed as a quad a ic unc ion:
V(x, ,
θ
θ
θ
)= TM +
θ
θ
θ
TM
θθθ
θ
θ
+2
θ
θ
θ
TM
θ
+2xTMT
+2xTMT
θ
θ
θ
θ
+xTM x(6)
whe e he ma ices can be ob ained om he sys em and he
con ol pa ame e s ([3]). Due o he con exi y p ope ies o
V(x, ,
θ
θ
θ
),p oblem(5)isequi alen o([3])
∗(x)=a g min
max
θ
θ
θ
∈ e (Θ
Θ
Θ)V(x, ,
θ
θ
θ
)
s. .Gxx+G ≤d
ε
(7)
whe e e (Θ
Θ
Θ)is he se o e ices o Θ
Θ
Θ.
The e minal egion Ωis assumed o sa is y he ollowing
condi ions:
•C1:I x∈Ω hen ACLx+D
θ
∈Ω, o e e y
θ
∈{
θ
∈
Rdim
θ
:∥
θ
∥∞≤
ε
}.
•C2:I x∈Ω hen u(x)=−Kx ∈U,whe eU!{u:
Fuu+Fxx≤g}.
Mo eo e , ma ix P ha cha ac e izes he e minal cos is
assumed o sa is y
•C3:P−AT
CLPACL >Q+KTRK.
The s abili y o ACL gua an ees he exis ence o a posi i e
de ini e ma ix Psa is ying C3.
The maximum cos o a gi en xand is deno ed as
V∗(x, )= max
θ
θ
θ
∈ e (Θ
Θ
Θ)V(x, ,
θ
θ
θ
)=V(x, ,0)(8)
+max
θ
θ
θ
∈ e (Θ)
θ
θ
θ
TH
θ
θ
θ
+2
θ
θ
θ
Tq(x, )
whe e H=M
θθ
,q(x, )=M
θ
+M
θ
xand V(x, ,0)=
TM +2xTMT
+xTM xis he pa o he cos ha does
no depend on he unce ain y. Wi h his de ini ion, p oblem
(7) can be ew i en as
∗(x)= a gmin
V∗(x, )
s. .Gxx+G ≤d
ε
,(9)
and he sys em is con olled by KMPC(x( )) = −Kx( )+
∗( | ),whe e ∗(x( )) = ! ∗( | )T,···, ∗( +N−1| )T"T.
III. A QP APPROACH TO MIN-MAX MPC
In his sec ion he main esul s o [1] a e p esen ed b ie ly.
In ha wo k, i is shown how he min-max p oblem (9) can
be eplaced by a ac able QP p oblem which p o ides a close
app oxima ion o he solu ion o he o iginal p oblem. This
can be accomplished wi h he ollowing s eps:
1) Ob ain an ini ial guess o he solu ion o (9), deno ed
˜ ∗.Asseenla e , hiscanbeachie edbysol ingaQP
p oblem.
2) Using ˜ ∗,ob ainaquad a ic unc iono ha bounds
he wo s case cos .
P ep in submi ed o 47 h IEEE Con e ence on Decision and Con ol.
Recei ed Ma ch 10, 2008.
3) Compu e he con ol law. This in ol es he solu ion o
aQPp oblem.
All hese s eps will be de ailed in he ollowing.
A. Compu ing ˜ ∗
Gi en Hde ined as in equa ion (8), deno e Ti=
∑N·dim
θ
j=1|Hij|,whe eHij deno es he (i,j)- h componen o
ma ix H.Then,de ine hediagonalma ixTas
T=diag(T1,···,Tn)(10)
Because o how ma ix Tis de ined, T−His a symme ic
diagonally dominan eal ma ix wi h nonnega i e diagonal
en ies, hus T−H≥0whichimplies ha T≥H.Le
˜
V(x, ,
θ
θ
θ
)be:
˜
V(x, ,
θ
θ
θ
)=V(x, ,0)+
θ
θ
θ
TT
θ
θ
θ
+2qT(x, )
θ
θ
θ
(11)
F om he inequali y T≥Hi is in e ed ha ˜
V(x, ,
θ
θ
θ
)≥
V(x, ,
θ
θ
θ
).Themaximumo ˜
V(x, ,
θ
θ
θ
)can be compu ed as
˜
V∗(x, )=max
θ
θ
θ
∈Θ
Θ
Θ
˜
V(x, ,
θ
θ
θ
)
=V(x, ,0)+ ace(T)
ε
2+2
ε
∥q(x, )∥1
=V(x, ,0)+∥H∥s
ε
2+2
ε
∥q(x, )∥1(12)
whe e ∥H∥sdeno es he sum o he absolu e alues o he
elemen s o H.Thenanini ialguesso hesolu iono (9)
can be ob ained as
˜ ∗(x)=a gmin
˜
˜
V∗(x,˜ )
s. .Gxx+G ˜ ≤d
ε
,(13)
This p oblem can be cas ed as a QP p oblem by making use
o slack a iables o deal wi h he 1-no m e m in ˜
V∗(x, ).
B. Ob aining an uppe bound o he wo s case cos
The uppe -bound o he maximum will be ob ained in wo
s eps. In he i s one we compu e a se o pa ame e s om
˜ ∗ ha allows us la e , in he second s ep, o compu e he
bound as a quad a ic unc ion o .
1) Compu ing he pa ame e ec o
α
( ):No e ha :
V∗(x, )= max
θ
θ
θ
∈ e (Θ
Θ
Θ)#
θ
θ
θ
1$T#Hq(x, )
qT(x, )V(x, ,0)$#
θ
θ
θ
1$
=max
∥z∥∞≤1
zTM( )z(14)
wi h
z=%
θ
θ
θ
T
ε
1
&T
,M( )=
#
ε
2H
ε
q(x, )
ε
qT(x, )V(x, ,0)$∈Rn×n,
whe e n=N·dim
θ
+1.
The ollowing p ocedu e p o ides an uppe bound o
he wo s case cos o a gi en .I compu es
α
( )=
[
α
1( ),...,
α
n−1( )]Tand a diagonal ma ix Γ( )≥M( )
such ha i s ace is an uppe bound o he wo s case cos
o (see p ope y 1 o [1]).
P ocedu e 1: Compu a ion o
α
( )=
[
α
1( ),...,
α
n−1( )]Tand Γ( ).
1) Le S(0=M( )∈Rn×n.
2) Fo k=1 on−1
3) Le M(k−1
sub =[S(k−1
ij ] o i,j=k···n.
4) Ob ain he pa i ion M(k−1
sub =#ab
T
bM
$,whe e
a∈R,b∈Rn−kand M ∈R(n−k)×(n−k).
5) Make
α
k( )='∥b∥1.
6) I
α
k( )=0 henS(k=S(k−1,elseS(k=S(k−1+
(0T
k−1,1
α
k( )−bT
α
k( ))T(0T
k−1,1
α
k( )−bT
α
k( )).
7) end o
8) Make Γ( )=S(n−1.
No e ha in he p e ious p ocedu e, 0m,ndeno es a (m×n)
ma ix o ze os. P ope y 1 o [1] shows ha he ace o
Γ( )cons i u es an imp o ed uppe bound o V∗(x, ).Tha
is, V∗(x, )≤ ace(Γ( )) ≤˜
V∗(x, ).
2) Ob aining he bound as a quad a ic unc ion on :
The diagonaliza ion p ocess shown in p ocedu e 1 can be
used o ob ain a ma ix deno ed by ˆ
Γ( ),whichallowsone
o ob ain a bound o he maximum ha can be compu ed as
aquad a ic unc iono .Thisisachie edbymeanso he
ollowing p ocedu e:
P ocedu e 2: Ob aining he ma ix ˆ
Γ( ).
1) Ob ain ˜ ∗ om he QP p oblem de ined in eq. (13).
2) Compu e
α
(˜ ∗)by p ocedu e 1.
3) Le ˆ
S(0( )=M( )∈Rn×n.
4) Fo k=1 on−1
5) Le ˆ
Msub( )=[ˆ
S(k−1
ij ( )] o i,j=k···n.
6) Ob ain he pa i ion ˆ
Msub( )=#a( )bT( )
b( )M ( )$,
whe e a( )∈R.
7) I
α
k(˜ ∗)=0 hen ˆ
S(k( )=
ˆ
S(k−1( ),else ˆ
S(k( )= ˆ
S(k−1( )+
(0T
k−1,1
α
k(˜ ∗)−b( )T
α
k(˜ ∗))T(0T
k−1,1
α
k(˜ ∗)−b( )T
α
k(˜ ∗)).
8) end o
9) Make ˆ
Γ( )= ˆ
S(n−1( ).
Deno e ha ˆ
V∗(x, )= ace (ˆ
Γ( )).Theo em1o [1]
shows ha ˆ
V∗(x, )is a quad a ic unc ion on and also
an uppe bound o he o iginal wo s case cos V∗(x, ).
C. Compu ing he con ol law
The alue o he con ol signal is ob ained by sol ing he
ollowing QP op imiza ion p oblem
ˆ ∗(x)=a gmin
ˆ
ˆ
V∗(x,ˆ )
s. .Gxx+G ˆ ≤d
ε
,(15)
and he sys em is con olled by ˆ
KMPC(x( )) = −Kx( )+
ˆ ∗( | ),whe eˆ ∗( | )is he i s elemen o ˆ
∗(x).
The compu a ional bu den o he p oposed s a egy is
much lowe han ha o he exac MMMPC. This com-
pu a ional bu den is mos ly due o he QP p oblems ha
mus be sol ed o ob ain he ini ial guess and he p oposed
solu ion i sel . No e ha in bo h cases, he complexi y o
each p oblem is he same as ha o a s anda d cons ained
MPC using a quad a ic cos unc ion. In con as o e alua e
he maximum cos V∗(x, )i is necessa y o e alua e he
unc ion o all he 2N∗dim
θ
e ices o Θ.No e ha hisisa
well known NP-ha d p oblem.
P ep in submi ed o 47 h IEEE Con e ence on Decision and Con ol.
Recei ed Ma ch 10, 2008.
Fig. 1. Pilo plan used o apply he MMMPC.
Hea exchange
Wa e ank
8
Fj
Fj
Tj,ou
Tj,in
F
F
TT2
T
Fig. 2. Diag am o he pilo plan wi h i s ou main elemen s: eac o ,
hea exchange , cooling jacke and al e.
IV. PROCESS DESCRIPTION
A ealp ocess ep esen edbyapilo plan hasbeenchosen
o he applica ion o he p oposed algo i hm. The p ocess
has been s udied p e iously by se e al au ho s [6], [17].
A. Labo a o y p ocess
The pilo plan (see Fig. 1) is used o simula e exo he mic
chemical eac ions based on empe a u e changes. I has
been used as a benchma k o con ol pu poses by se e al
esea che s [13], [6]. The main elemen s o he pilo plan
a e he eac o , he hea exchange , he cooling jacke and he
al e o manipula e he low a e h ough he cooling jacke
(see Fig. 2).
Acoolingjacke isused o educe he eac o empe a u e.
The hea dissipa ion can be egula ed by he al e 8which
manipula es he low a e Fj h ough he cooling jacke .
The cooling luid, wa e , en e s he cooling jacke wi h a
cons an empe a u e. The eac i e is supplied o he eac o
by he eed F ,in o keep he chemical eac ion ac i e. Be o e
en e ing he eac o , he eed passes h ough a hea exchange
in o de o adop he empe a u e o he eac o con en .
The ou low F ,ou is used o keep he olume o he eac o
con en cons an .
To simula e exo he mic eac ions, he eac o possesses an
elec ical esis ance in o de o supply calo ic ene gy. The
ene gy o be supplied by he 14.4kW elec ical esis ance
is calcula ed by means o a ma hema ical model o he
simula ed eac ion. The use o a esis ance means ha no
chemical eac ion akes place in he eac o , ins ead he
eac ion is emula ed on basis o empe a u e changes, as done
in [15].
B. Ma hema ical model
Al hough i is no necessa y o ha e a ma hema ical model
o he design o he min-max p edic i e con olle , his
sec ion shows he p ocess model o emphasize i s nonlinea
cha ac e . The ma hema ical model also jus i ies he way o
emula e he hea gene a ed by he chemical eac ion wi h he
aid o he esis ance.
The emula ed chemical eac ion, ep esen ing a e inemen
p ocess, was used p e iously in [6]. Wi h F =F ,in =F ,ou
and a cons an olume, he model o he chemical eac ion
can be de ined as:
dT
d =−Fj
V(Tj,in −Tj,ou )
+(−∆H)·V
MCp
k0e−E/(RT)C2
A(16)
dCA
d =F
V(CA,in−CA)−k0e−E/(RT)C2
A(17)
deno ing Fj,Tj,in and Tj,ou he low a e h ough he jacke
and he empe a u e o he wa e en e ing and lea ing he
cooling jacke , espec i ely. CAand CA,in ep esen he eac-
i e concen a ion in he eac o and in he eed, espec i ely.
The eed passes h ough he hea exchange and en e s he
eac o nea ly wi h he empe a u e o he eac o con en .
Thus i is assumed ha no hea is nei he emo ed o supplied
due o he eed. The hea exchange in he cooling jacke is
gi en by he ollowing empi ical model:
Fj·(Tj,ou −Tj,in)=T−
α
β
(1−e−
γ
Fj)(18)
wi h
α
=292.19K,
β
=14.94s/land
γ
=13.18s/l.
The chemical eac ion is nonlinea in he dynamics o he
empe a u e and he concen a ion due o he quad a ic e ms
o he concen a ion in he model equa ions (16) and (17).
Fo u he de ails on he model pa ame e s see [6].
V. EXPERIMENTAL RESULTS
The s a egy desc ibed in sec ion III has been applied
o he e inemen p ocess. In his sec ion he expe imen al
esul s will be exposed and discussed. CARIMA ype p edic-
ion models wi h bounded addi i e unce ain ies we e used in
he expe imen s. This ype o model ex ends he concep o
noise in adi ional CARIMA models so ha an unce ain y
is conside ed:
A(z−1)y( )=z−dB(z−1)u( −1)+C(z−1)
θ
( )
∆(19)
P ep in submi ed o 47 h IEEE Con e ence on Decision and Con ol.
Recei ed Ma ch 10, 2008.
wi h ∆=1−z−1,
θ
( )∈{
θ
∈Rdimy :∥
θ
∥∞≤
ε
},anddimy
he dimension o y( ).Theuseo his ypeo p edic ion
models esul s in a con ol law wi hou e o in s eady
s a e. The e a e ew di e ences be ween implemen ing he
algo i hm o sec ion III o a s a e space model and a
CARIMA model wi h bounded addi i e unce ain ies. The
main di e ence is he me hod used o ind he ma ices o
he p edic ion equa ion [3]. The cos unc ion is he same as
in (6). Appendix A p esen s p ac ical aspec s ela ed o he
use o a CARIMA model (19) o p edic ion.
In he ollowing sec ions he con ol sys em in he pilo
plan will be desc ibed and he necessa y s eps o ob ain
ap edic ionmodelwillbep esen ed.Finally,expe imen al
esul s will be exposed.
A. Desc ip ion o he con ol sys em
The senso s and ac ua o s in he plan a e connec ed
o a PMC-10 con ol uni . The PMC-10 is connec ed by
ARCne o a pe sonal compu e ha uns he con ol and
moni o ing sys em Sima ic-IT.Thecon olalgo i hmhas
been implemen ed di ec ly in Ma lab and he communica ion
wi h Sima ic-IT is done using he OPC p o ocol (OLE o
P ocess Con ol). Bo h Sima ic-IT and he con olle un
on he same pe sonal compu e , based on a Pen ium II
p ocesso a 300 Mhz. This compu e does no ha e enough
compu a ional powe o sol e exac ly he min-max p oblem
o a ypical MMMPC, bu can compu e he con ol ac ion
using he p oposed s a egy.
B. Iden i ica ion o he p edic ion model
APRMSS (Pseudo-Random Mul ile el S ep Sequence)has
been applied o he eci cula ion al e wi h he objec i e
o collec ing da a o he pa ame e iden i ica ion o he
p edic ion model. The pe iods o he PRMSS ha e been
chosen su icien ly long o obse e he eac ion o he pilo
plan o changes in he inpu (see Fig. 3). I can be seen ha
he empe a u e o he ank eaches s eady s a e in each s ep
in some hing mo e han wo hou s, al hough he a ia ions in
s eady s a e a e o se e al deg ees. The eagen concen a ion
also su e s a ia ions in s eady s a e. I can be obse ed ha
he inpu –ou pu gain is nega i e and clea ly a iable (g ea e
gain o low openings o 8). A i s o de ans e unc ion
model wi h delay is p oposed as p edic ion model. This low
o de model canno co ec ly desc ibe he dynamics o he
plan , bu i is a good app oach o check he obus ness o
he con olle in p esence o unce ain ies and dis u bances.
Using he da a o Fig. 3 he ollowing model has been
iden i ied:
G(s)= −0.975
950s+1e−31.25s(20)
This model was disc e ized wi h a sampling ime o Ts=60s.
The delay was ounded o 1 sampling ime in o de o a oid
app oxima ions o he ime delay, e.g. Pad´e app oxima ion.
The eby, he ollowing CARIMA model was ob ained:
y( +1)=0.939y( )−0.0597u( −1)+
θ
( )
∆(21)
wi h he noise polynomial C(z−1)=1.
8[%]
CA[mol/l]
T[oC]
[min]
0
0
0
100
100
100
200
200
200
300
300
300
400
400
400
500
500
500
600
600
600
30
40
40
50
60
60
70
80
80
0
0.1
0.2
0.3
0.4
Fig. 3. Expe imen o he p edic ion model iden i ica ion. F om op o
bo om: Tank empe a u e (T), al e opening ( 8)y eagen concen a ion
(CA).
e o
[min]
100 200 300 400 500 600
−0.5
−0.25
0
0
0.25
0.5
Fig. 4. One s ep ahead p edic ion e o du ing he expe imen o he
model iden i ica ion.
C. Expe imen al esul s o he con olle
The p oposed con ol s a egy was applied o he pilo
plan desc ibed in sec ion IV-A using (21) as a p edic ion
model. Fo he p edic ion and con ol ho izons alues o N=
15 and Nu=12 ha e been used1.The e o e hep edic ion
ho izons includes app oxima ely one ime cons an o he
p ocess, a common alue o his pa ame e in p edic i e
con ol. The weigh ing ac o o he con ol e o has been
chosen equal o Rj=2. Based upon he one s ep ahead
p edic ion e o (see Fig. 4) he pa ame e
ε
has been chosen
o
ε
=0.25. As a esul , in 97% o he samples he one
s ep ahead p edic ion e o is bounded by he chosen alue.
Finally, in o de o es ic he sys em inpu and ou pu in
he expe imen s, he ollowing cons ain s ha e been used:
30 ≤ˆy( +j| )≤70,j=2,...,16,∀
θ
θ
θ
∈ e (Θ
Θ
Θ)
5≤u( +j| )≤100,j=0,...,11
−20 ≤∆u( +j| )≤20,j=0,...,11
No e ha in he ou pu es ic ions he e ec o he unce -
ain y has o be conside ed.
In o de o analyse he sys em beha iou , se e al ex-
pe imen s wi h e e ence changes and dis u bance ejec ion
ha e been made using he p oposed con ol s a egy. Fig.
5shows he esul so he ackingexpe imen wi h e -
e ences di e en enough o esul in con ol ac ions in a
la ge in e al. A e he i s e e ence change no o e shoo
1The model delay implies ha he p edic ion ho izon s a s in +2and
ends in +16.
P ep in submi ed o 47 h IEEE Con e ence on Decision and Con ol.
Recei ed Ma ch 10, 2008.

8[%]T[oC]
mol/
[min]
0
0
150
150
35
40
40
45
50
50
50
55
60
60
65
20
80
100
100
100
TT2[oC]CA[mol/l]
[min]
0
0
150
150
50
50
100
100
0.1
0.15
0.2
0.25
0.3
17
17.5
18
18.5
Fig. 5. Re e ence acking expe imen . F om op o bo om: Tank
empe a u e (T), al e opening ( 8), eagen concen a ion (CA)andcold
wa e empe a u e (TT2).
appea s in spi e o a qui e as con olle eac ion. A e
he second e e ence change a small o e shoo (o abou
−0.5oC), jus i ied by he nonlinea p ocess beha iou , can be
obse ed. In s eady s a e he con olle shows small changes
in he con ol ac ion necessa y o s abilize he ou pu on he
e e ence in p esence o a ia ions in he gene a ed hea and
he cold wa e empe a u e.
In second place a dis u bance ejec ion expe imen was
ca ied ou . The esul s o a dis u bance in he sys em
inpu , he opening o he al e 8,a ep esen edinFig.
6. As can be seen, a e app oxima ely 70min a cons an
dis u bance in he inpu o ∆ 8=15% was applied. The
con olle eac s apidly and ejec s he pe u ba ion in abou
20 minu es. A e he disappea ance o he pe u ba ion in =
101min he con olled sys em shows he same beha iou and
eaches s eady s a e in app oxima ely 20 minu es. Nei he he
empe a u e no he con ol ac ion show oscilla ions a e he
pe u ba ion.
The hi d expe imen , using an addi i e dis u bance in he
eeding F ,isshowninFig.7.In =70min a change in
he eeding low o ∆F =0.0125l/s, which co esponds o
an e o o 25%, has been applied. Wi h an inc easing e o ,
he con olle educes he opening o he al e and eaches
acompensa iono hedi e gencea e 15minu es.In his
expe imen an o e shoo o −0.50oCcanbeobse ed.The
oscilla ion in he empe a u e and he con ol ac ion is qui e
small and seems accep able due o he s ong dis u bance.
The e e ence acking expe imen was epea ed wi h a
linea cons ained p edic i e con olle (GPC) o allow he
compa ison be ween he p oposed s a egy and a s anda d
MPC me hod. The GPC is based on he linea model (21)
8[%]T[oC]
[min]
0
0
120
120
40
40
40
45
50
50
55
60
60
60
60
65
70
20
20
80
80
100
100
30
u[%]
TT2[oC]CA[mol/l]
[min]
0
0
0
120
120
120
40
40
40
40
50
60
60
60
60
70
20
20
20
80
80
80
100
100
100
0.05
0.1
0.15
0.2
17
17.5
18
18.5
30
Fig. 6. Expe imen wi h inpu dis u bance ejec ion. F om op o bo om:
Tank empe a u e (T), al e opening( 8), con olle ou pu (u), eagen
concen a ion (CA)andcoldwa e empe a u e(TT2).
8[%]T[oC]
[min]
0
0
40
40
40 40
45
50
50
55
60
60
60
60
65
70
20
20
80
80
80
100
100
30
TT2[oC]CA[mol/l]
[min]
0
0
40
40
60
60
20
20
80
80
100
100
0.05
0.1
0.15
0.2
0.25
17
17.5
18
18.5
Fig. 7. Expe imen wi h dis u bance ejec ion in he eed low. F om op
o bo om: Tank empe a u e (T), al e opening( 8), eagen concen a ion
(CA)andcoldwa e empe a u e(TT2).
P ep in submi ed o 47 h IEEE Con e ence on Decision and Con ol.
Recei ed Ma ch 10, 2008.
MMMPC
MPC
e
MMMPC
MPC
8[%]T[oC]
[min]
[min]
20
80
100
35
40
40
45
50
55
60
60
65
0
0
25
25
50
50
75
75
100
100
125
125
150
150
Fig. 8. Re e ence acking esul s o he MMMPC and he GPC. F om op
o bo om: Tank empe a u e (T), al e opening ( 8).
and was used wi h he same pa ame e s as he MMMPC. I
can be obse ed in he esul s (see Fig. 8) ha he p ocess
con olled by he GPC exhibi s signi ican oscilla ions in
he empe a u e and he con ol ac ion a e he e e ence
changes. The compa ison o he esul s shows ha he
MMMPC s abilizes he empe a u e mo e e icien ly and wi h
less oscilla ions in he opening o he al e.
Finally, i is impo an o men ion ha he calcula ion o
he con ol signal ook place wi hou p oblems wi hin he
chosen sampling ime (60 seconds). Du ing he expe imen s
he a e age compu a ion ime was 5.64 seconds, wi h a
maximum o 9.90 seconds and a minimum o 1.863 seconds.
VI. CONCLUSIONS
In his pape an MMMPC based on an ac able QP p ob-
lem was applied o a pilo plan . The esul s showed a good
sys em beha iou and he s abilisa ion o he plan empe-
a u e a ound he ope a ion poin . A e e e ence changes he
con olle quickly compensa es de ia ions. Fu he mo e, he
MMMPC showed i s capaci y o compensa e e o s caused
by he dis u bances.
The applica ion o a p ocess shown in his wo k joins
he small numbe o MMMPC applica ions epo ed in
specialised li e a u e. The low compu a ional equi emen s o
he p oposed con ol s a egy allowed he use o app op ia e
sampling imes and ealis ic p edic ion and con ol ho izons.
The eby i is shown ha he use o p oposed s a egy allows
he applica ion o his kind o con olle s o a la ge numbe
o p ocesses.
REFERENCES
[1] T. Alamo, D.R. Ram´ı ez, D. Mu˜noz de la Pe˜na, and E.F. Camacho.
Min-max MPC using a ac able QP p oblem. Au oma ica,43:693–
700, 2007.
[2] A. Bempo ad, F. Bo elli, and M. Mo a i. Min-max Con ol o Con-
s ained Unce ain Disc e e-Time Linea Sys ems. IEEE T ansac ions
on Au oma ic Con ol,48(9):1600–1606,2003.
[3] E.F. Camacho and C. Bo d´ons. Model P edic i e Con ol.Sp inge -
Ve lag, second edi ion, 2004.
[4] P.J. Campo and M. Mo a i. Robus Model P edic i e Con ol. In P oc.
Ame ican Con ol Con e ence,pages1021–1026,June10-121987.
[5] D. Mu˜noz de la Pe˜na, D.R. Ram´ı ez, E.F. Camacho, and T. ´
Alamo.
Applica ion o an explici min-max mpc o a scaled labo a o yp ocess.
Con ol Enginee ing P ac ice,13:1463–1471,2005.
[6] J.K. G ube and C. Bo dons. Con ol P edic i o no Lineal Basado
en Modelos de Vol e a. Aplicaci´on a una Plan a Pilo o. Re is a
Ibe oame icana de Au om´a ica e In o m´a ica Indus ial (in spanish),
4(3):34–45, 2007.
[7] E.C. Ke igan and J.M. Maciejowski. Feedback min-max Model
p edic i e Con ol Using a Single Linea P og am: Robus S abili y
and he Explici Solu ion. In e na ional Jou nal o Robus Nonlinea
Con ol,14:395–413,2004.
[8] Y.H. Kim and W.H. Kwon. An Applica ion o Min-Max Gene al-
ized P edic i e Con ol o Sin e ing P ocesses. Con ol Enginee ing
P ac ice,6:999–1007,1998.
[9] M.V. Ko ha e, V. Balak ishnan, and M. Mo a i. Robus cons ained
model p edic i e con ol using linea model inequali ies. Au oma ica,
32(10):1361–1379, 1996.
[10] J.H. Lee and Zhenghong Yu. Wo s -case o mula ions o model
p edic i e con ol o sys ems wi h bounded pa ame e s. Au oma ica,
33(5):763–781, 1997.
[11] Y. Lu and Y. A kun. Quasi-Min-Max MPC Algo i hms o LPV
sys ems. Au oma ica,36(4):527–540,2000.
[12] D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokae . Con-
s ained model p edic i e con ol: S abili y and op imali y. Au oma ica,
36:789–814, 2000.
[13] D.R. Ram´ı ez, M.R. A ahal, and E.F. Camacho. Min-Max P edic i e
Con ol o a Hea Exchange using a Neu al Ne wo k Sol e . IEEE
T ans. on Con ol Sys ems Technology,12(5):776–786,2004.
[14] D.R. Rami ez and E.F. Camacho. Piecewise A ini y o Min-Max
MPC wi h bounded addi i e unce ain ies and a quad a ic c i e ion.
Au oma ica,42:295–302,2006.
[15] L.O. San os, P.A.F.N.A. A onso, J.A.A.M. Cas o, N.M.C. Oli ei a,
and L.T. Biegle . On-line implemen a ion o nonlinea MPC: an
expe imen al case s udy. Con ol Enginee ing P ac ice,9(8):847–857,
2001.
[16] P.O.M. Scokae and D.Q. Mayne. Min-max eedback model p e-
dic i e con ol o cons ained linea sys ems. IEEE T ansac ions on
Au oma ic Con ol,43(8):1136–1142,1998.
[17] F. Szei e , T. Cho an, and L. Nagy. P ocess dynamics and empe a u e
con ol o ed-ba ch eac o s. Compu e s & Chemical Enginee ing,
19(1):447–452, 1995.
APPENDIX A. PRACTICAL ASPECTS RELATED TO THE
INPUT/OUTPUT DESCRIPTION
This appendix p esen s he main di e ences be ween he
implemen a ion o he p oposed s a egy wi h a s a e space
model (1) and an inpu /ou pu model like (19). In his case,
he e olu ion o he ou pu along he p edic ion ho izon can
be desc ibed in condensed o m as [3]:
y=Guu+G
θθ
θ
θ
+Fxx(22)
The ma ices Gu,G
θ
and Fxcan be ob ained om he o iginal
model using se e al di e en me hods, e.g. he Diophan ine
equa ion [3]. A mo e in ui i e me hod is he use o he
s ep esponse coe icien s [3] o o m ma ix Gu.The ee
esponse ec o Fxxcan easily be compu ed by i e a ing wi h
he p ocess model wi h he assump ion o a cons an con ol
signal along he p edic ion ho izon. Ma ix G
θ
is compu ed
by ea ing he unce ain y as an addi ional sys em inpu
weigh ed by a uni polynomial in he model. Thus, using he
me hod o he Diophan ine equa ion, G
θ
can be compu ed as
Gu,bu assuming ha B(z−1)=1. Also, G
θ
can be o med
om he coe icien s o he sys em esponse o a s ep in he
unce ain y
θ
( ).The ec o ucon ains he u u e con ol
inc emen s i a CARIMA model is used [3]. In his case
s a e ec o xcon ains he p esen and pas ou pu alues as
P ep in submi ed o 47 h IEEE Con e ence on Decision and Con ol.
Recei ed Ma ch 10, 2008.
well as he pas con ol inc emen s:
x=[y( ),...,y( −na),∆u( −1),...,∆u( −nb)]T
deno ing naand nb he polynomial o de s o (1−z−1)A(z−1)
and B(z−1), espec i ely.
The di e en model implies changes in he cos unc ion,
bu , as shown in he ollowing, i can be ew i en as in (6).
The usual o m o he cos unc ion wi h a p edic ion model
like (19) is:
V(x,u,
θ
θ
θ
)=
N
∑
j=1
e( +j| )TQe( +j| )+
Nu−1
∑
j=0
∆u( +j)TRm∆u( +j)(23)
being e( +j| ) he p edic ed e e ence acking e o a ime
+j:
e( +j| )=(y( +j| )−w( +j))
whe e w( +j)is he e e ence a ime +j.Wi h he
p edic ion equa ion (22) and he assump ion Q=1 hecos
unc ion can be w i en as:
V(x,u,
θ
θ
θ
)=(Guu+G
θθ
θ
θ
+Fxx−w)T
·(Guu+G
θθ
θ
θ
+Fxx−w)
+uTRu(24)
wi h Radiagonalma ixo he o m:
R=diag(Rm,...,Rm)(25)
The ec o wcon ains he u u e alues o he e e ence
ajec o y and is de ined as w=[w( +1),...,w( +N)]T.
I a ze o e e ence is used, i.e. w( +j)=0 o j=1,...,N,
he cos unc ion can be exp essed as in (6) being he
ma ices Muu =GT
uGu+R,M
θθ
=GT
θ
G
θ
,M
θ
u=GT
θ
Gu,
Mu =GT
uFx,M
θ
=GT
θ
Fxand M =FT
xFx.In hecaseo
anonze o e e ence, hecompu a iono hecos unc ion
can be ca ied ou in an analogous way and does no change
he op imisa ion algo i hm. F om his poin , he p oposed
s a egy can be applied in he same way as wi h s a e space
models.
P ep in submi ed o 47 h IEEE Con e ence on Decision and Con ol.
Recei ed Ma ch 10, 2008.
View publica ion s a sView publica ion s a s