Enla ging hedomaino a ac iono MPCcon olle s
D. Limon, T. Alamo and E.F. Camacho
Depa amen o de Ingenie ´
ıa de Sis emas y Au om´
a ica, Uni e sidad de Se illa.
Escuela Supe io de Ingenie os, Camino de los Descub imien os s/n. 41092 Se illa, SPAIN.
Telephone: +34 954487357 Fax: +34 954487340
Abs ac
This pape p esen s a me hod o enla ging he domain o a ac ion o nonlinea model p edic i e con ol (MPC). The usual way o
gua an eeing s abili y o nonlinea MPC is o add a e minal cons ain and a e minal cos o he op imiza ion p oblem such ha he
e minal egion is a posi i ely in a ian se o he sys em and he e minal cos is an associa ed Lyapuno unc ion. The domain o
a ac ion o he con olle depends on he size o he e minal egion and he con ol ho izon. By inc easing he con ol ho izon, he
domain o a ac ion is enla ged bu a he expense o a g ea e compu a ional bu den, while inc easing he e minal egion p oduces an
enla gemen wi hou an ex a cos .
In his pape , he MPC o mula ion wi h e minal cos and cons ain is modi ied, eplacing he e minal cons ain by a con ac i e
e minal cons ain . This cons ain is gi en by a sequence o se s compu ed o -line ha is based on he posi i ely in a ian se . Each se
o his sequence does no need o be an in a ian se and can be compu ed by a p ocedu e which p o ides an inne app oxima ion o he
one-s ep se . This p ope y allows us o use one-s ep app oxima ions wi h a ade o be ween accu acy and compu a ional bu den o he
compu a ion o he sequence. This s a egy gua an ees closed loop s abili y ensu ing he enla gemen o he domain o a ac ion and he
local op imali y o he con olle . Mo eo e , his idea can be di ec ly ansla ed o obus MPC.
Key wo ds: Model P edic i e Con ol, cons ained nonlinea sys ems, domain o a ac ion, in a ian se s, s abili y.
1 In oduc ion
One o he main ac o s o he success o MPC bo h in indus-
y and academia is he ease wi h which i inco po a es con-
s ain s in bo h he s a es and he inpu s o he sys em. Fu -
he mo e, a heo e ical amewo k o analyzing such opics
as s abili y, obus ness, op imali y, e c. o nonlinea sys ems
has ecen ly been de eloped: see (Mayne, Rawlings, Rao &
Scokae 2000) o a su ey, o (Camacho & Bo dons 1999)
o p ocess indus y applica ion issues.
One o he mos impo an esul s in he s abili y analysis
o MPC is he addi ion o a e minal cons ain based on
an in a ian se (Michalska & Mayne 1993). This echnique
imp o es p e ious e minal equali y cons ain esul s, bu
equi es commu a ion o a local con olle when he s a e
eaches he e minal egion. This p oblem is o e come by
?A p elimina y e sion o his pape was p esen ed a IFAC Wo ld
Cong ess 2002 (Ba celona, Spain).
The au ho s would like o acknowledge MCYT-Spain (con ac
DPI2002-04375-c03-01) o unding his wo k and also o hank
he anonymous e iewe s o hei help ul commen s.
Email add ess: {
limon, alamo, edua do
}
@ca uja.us.es
(D. Limon, T. Alamo and E.F. Camacho).
adding a e minal cos o he unc ional o be op imized
(Chen & Allg¨
owe 1998, Mayne e al. 2000).
The domain o a ac ion o he MPC con olle is he se
o s a es which can be s ee ed o he e minal egion in N
s eps o less, whe e Nis he con ol ho izon. The size o
he domain o a ac ion depends on he size o he e mi-
nal egion and he chosen con ol ho izon. Inc easing bo h
o hem yields a bigge domain o a ac ion. The mos used
p ocedu e o enla ge he domain o a ac ion is o inc ease
he p edic ion ho izon N. This leads o a g ea e numbe o
decision a iables and, he e o e, o a g ea e compu a ional
e o . Howe e , enla ging he size o he e minal se p o-
ides a la ge domain o a ac ion wi h he same compu a-
ional cos .
The enla gemen o he e minal se has been used o
cons ained linea sys ems in (De Don´
a, Se on, Mayne
& Goodwin 2002, Limon, Gomes da Sil a, Alamo &
Camacho 2003), whe e he sa u a ed local con ol law has
been conside ed. In (Chen, Ballance & O’Reilly 2001)
he e minal se is enla ged by using a local LDI ep e-
sen a ion o he nonlinea sys em and by sol ing o -line
an LMI op imiza ion p oblem. In (Cannon, Deshmukh &
Kou a i akis 2003), a local LDI ep esen a ion is also used,
P ep in submi ed o Au oma ica 14 Oc obe 2004
and a poly opic e minal se and an associa ed e minal
cos a e compu ed. In (Magni, De Nicolao, Magnani &
Sca olini 2001), he enla gemen o he domain o a ac-
ion is achie ed by conside ing a p edic ion ho izon la ge
han he con ol ho izon.
This pape p esen s a me hod o enla ge he domain o a ac-
ion o MPC by inc easing he size o he e minal egion. I
is achie ed by a new idea: eplacing he e minal cons ain
by a con ac i e cons ain gi en by a sequence o eachable
se s o a gi en in a ian se . This is a sequence o se s (no
necessa ily in a ian ) whe e he sys em can be admissibly
s ee ed om one se o he ollowing, ul ima ely eaching
he a ge in a ian se . This sequence o se s is compu ed
o line by ecu sion based on he posi i ely in a ian se . I
is shown ha his sequence can be compu ed using an inne
app oxima ion o he one s ep se o elax he compu a ional
bu den o exac compu a ion. The p oposed con olle gua -
an ees he enla gemen o he domain o a ac ion, asymp-
o ic s abili y and local op imali y o he closed loop sys-
em. Fu he mo e, i can be di ec ly ansla ed o he obus
MPC o mula ion by using a sequence o obus ly eachable
se s. I is wo h ema king ha he op imiza ion p oblem,
and hence he on line compu a ional e o , o he p oposed
MPC is simila o he o iginal one.
2 Sys em desc ip ion
Conside asys emdesc ibedby a nonlinea in a ian disc e e
ime model
x+= (x,u)(1)
whe e x∈IRnis he sys em s a e, u∈IRmis he cu en
con ol ec o and x+is he successo s a e. The sys em is
subjec o cons ain s on bo h s a es and con ol ac ions, and
hey a e gi en by
x∈X(2)
u∈U(3)
whe e Xis a closed se and Ua compac se , bo h o hem
con aining he o igin.
Conside a sequence o con ol ac ions u o be applied o
he sys em a cu en s a e x. Then, he p edic ed s a e o
he sys em a ime j, i he ini ial s a e is x(a ime 0)
and he con ol sequence uis applied, will be deno ed as
x(j) = ϕ(j;x,u).
3 Compu a ion o a sequence o eachable se s.
In he ollowing some well es ablished de ini ions and e-
sul s on in a iance se heo y (see (Blanchini 1999)) a e p e-
sen ed:
Conside an au onomous sys em x+= (x), hen he se
Ω⊂IRnis a posi i ely in a ian se i (x)∈Ω, o all
x∈Ω. A se Ω⊂IRnis a con ol in a ian se o he sys em
(1) subjec o cons ain (3) i o all x∈Ω, he e exis s
an admissible inpu u=u(x)∈Usuch ha (x,u)∈Ω.
Le Ω⊂IRnbe a posi i ely (o con ol) in a ian se o a
sys em (1) subjec o cons ain (2) and (3), hen he i-s ep
s abilizable se Xi(Ω)is he se o admissible s a es which
can be s ee ed o he a ge se Ωin is eps o less by a
sequence o admissible con ol ac ions.
A in e es ing de ini ion in in a ian se heo y is he so-
called one-s ep se : le Ω⊂IRn, hen he one-s ep se o
Ω,Q(Ω), o he sys em (1) subjec o (3), is he se o
s a es which can be s ee ed in one s ep o he a ge se
Ωby an admissible con ol ac ion, i.e. Q(Ω) = {x∈IRn:
∃u(x)∈Usuch ha (x,u)∈Ω}. I he sys em is con olled
by u=h(x), he closed loop sys em is cons ained o he
admissible se Xh={x∈X:h(x)∈U}and he closed-loop
one-s ep se is gi en by Qh(Ω) = {x∈Xh: (x,h(x)) ∈Ω}.
I is easy o see ha Qh(Ω)⊆Q(Ω).
This se ope a ion allows us o claim ha a gi en se Ωis
a con ol in a ian se i and only i Ω⊆Q(Ω). Mo eo e ,
he one s ep se has he ollowing p ope ies: a) i Ω1⊆Ω2,
hen Q(Ω1)⊆Q(Ω2)and b) Q(Ω1∪Ω2) = Q(Ω1)∪Q(Ω2).
In he ollowing lemma, some in e es ing p ope ies o he
i-s ep s abilizable se a e gi en.
Lemma 1 Conside X0(Ω) = Ω⊆X, hen
(i) Xi(Ω) = Q(Xi−1(Ω))∩X, o i ≥1.
(ii) Xi(Ω)⊇Xi−1(Ω)and Xi(Ω)is a con ol in a ian se .
(iii) Xi(Xj(Ω)) = Xi+j(Ω).
(i ) Xi(Ω1∪Ω2) = Xi(Ω1)∪Xi(Ω2).
3.1 Ob aining a sequence o eachable se s.
The objec i e o his sec ion is o p esen a gene al and p ac-
ical p ocedu e o compu e a con ac i e sequence o each-
able se s, {Ωi}, based on he e minal se Ω. We deno e as
sequence o eachable se s a sequence o se s whe e he sys-
em s a e can be s ee ed om one se Ωi o he ollowing,
Ωi−1, in an admissible way, inally eaching he a ge in a i-
an se Ω. This p oblem has been s udied in (Be sekas 1971)
whe e i is demons a ed ha he maximal sequence ha can
be ob ained is he s abilizable se Xi(Ω). The compu a ion o
his sequence is based on he calcula ion o he one-s ep se .
The compu a ion o in a ian se s, and hence o he one-s ep
se , is an open ield (see (Blanchini 1999) o a compila ion
o he exis ing esul s). E icien p ocedu es exis o com-
pu e i o linea sys ems subjec o poly opic cons ain s,
o sys ems wi h poly opic cons ain s desc ibed by linea
di e en ial inclusions (Blanchini 1999). Howe e , o non-
linea sys ems he e is no a gene al p ocedu e o his.
2
In o de o elax he complexi y o compu a ion, he one-
s ep se can be eplaced by an inne app oxima ion o i ,
i.e. Qap(Ω)⊆Q(Ω). This elaxa ion makes sense o he
sake o he ac abili y o he p ocedu e used o compu e i .
Using Qap(·), and based on he in a ian se Ω, a con ac i e
sequence o eachable se s can be compu ed by he ollowing
ecu sion:
Ωi=Qap(Ωi−1)∩X,wi h Ω0=Ω(4)
This sequence o se s has he ollowing p ope ies:
Lemma 2 Le {Ωi}be a sequence o se s ob ained by (4),
hen
(i) Ωi⊆Xi(Ω). In ac , i Qap(·) = Q(·), hen Ωi=Xi(Ω).
(ii) I Ωi−1⊆Ωi hen Ωiand Ωi−1a e con ol in a ian
se s.
(iii) XN−1(Ωi)⊆XN(Ωi−1), o all N ≥1and i ≥1.
P oo :
(i) Ω1=Qap(Ω)∩X⊆Q(Ω)∩X=X1(Ω). Conside
ha Ωi−1⊆Xi−1(Ω), hen Ωi=Qap(Ωi−1)∩X⊆
Q(Ωi−1)∩X⊆Q(Xi−1(Ω))∩X=Xi(Ω).
(ii) Ωi−1⊆Ωi=Qap(Ωi−1)∩X⊆Q(Ωi−1)⊆Q(Ωi), and
he p oo is de i ed om he geome ic condi ion o
in a iance.
(iii) The compu ed sequence sa is ies ha Ωi⊆Q(Ωi−1)∩
X=X1(Ωi−1). Then XN−1(Ωi)⊆XN−1(X1(Ωi−1)) =
XN(Ωi−1).2
No e ha he ob ained sequence inhe i s some p ope ies
om he s abilizable se s, bu , i is no gua an eed ha Ωi
includes ei he he se Ωi−1o Ω, gi en he app oxima e
cha ac e o Qap(·). Consequen ly, he ob ained sequence is
a sequence o eachable se s (no necessa ily in a ian se s)
o he a ge se Ω. This esul allows us o design algo i hms
less compu a ionally demanding o de e mining a sequence
o in a ian s se s o me ely eachable se s. Simila ideas ha e
been used o he compu a ion o posi i ely in a ian se s
o nonlinea sys ems based on an LDI app oxima ion o he
sys em by sol ing an LMI (Chen e al. 2001). In (Cannon
e al. 2003), using an LDI ep esen a ion o he nonlinea
sys ems, a sequence o poly opic in a ian se s is compu ed
and an in e pola ion based con olle is p oposed. An algo-
i hm o compu ing a poly opic se Qap(Ω) o nonlinea
sys ems based on in e al a i hme ics is p esen ed in (B a o,
Limon, Alamo & Camacho 2003). The app oxima ion can
be ob ained wi h a gi en bound on he e o , which allows
a ade o be ween he accu acy o he app oxima ion and
he compu a ional bu den o be ound.
4 The MPC echnique
MPC is a well es ablished con ol s a egy capable o ob-
aining an op imal con ol law ha akes in o accoun con-
s ain s on he s a e and on he con ol ac ions. Mo eo e ,
unde mild assump ions, i is possible o gua an ee closed
loop asymp o ic s abili y (Mayne e al. 2000). The con ol
law KN(x)is ob ained by sol ing he ollowing cons ained
op imiza ion p oblem
min
uVN(x,u) =
N−1
∑
i=0
`(x(i),u(i))+F(x(N))
s. .x(i)∈X,u(i)∈U,i=0,···,N−1
x(N)∈Ω
whe e x(i) = ϕ(i;x,u), and applying he op imal solu ion o
he sys em in a eceding ho izon way. This ini e ho izon
nominal MPC op imiza ion p oblem wi h e minal cos and
e minal cons ain is he mos gene al way o o mula ing
he MPC con olle , and in he ollowing his o mula ion
will be deno ed as s anda d MPC. Taking in o accoun ha
he op imal minimize u∗(x)only depends on he ac ual s a e
xand he eceding ho izon policy, he con ol law is gi en
by u=KN(x) = u∗(0). This con ol law s abilizes he sys em
asymp o ically unde he ollowing assump ions:
Theo em 3 (Mayne e al. 2000) Le u =h(x)be a con ol
law such ha Ω⊆Xh={x∈X:h(x)∈U}is a posi i ely
in a ian se o he closed loop sys em. Le F(x)be a Lya-
puno unc ion associa ed o he sys em in Ω, such ha o
all x ∈Ω, F( (x,h(x)))−F(x)≤ −`(x,h(x)) hen, he MPC
con ol law s abilizes he sys em asymp o ically o all ini-
ial s a es such ha he op imiza ion p oblem is easible.
Unde hese assump ions, he op imal cos unc ion V∗
N(x)is
a Lyapuno unc ion o he closed loop sys em and i s do-
main o a ac ion is he N-s ep s abilizable se o he e mi-
nal egion Ω,XN(Ω). The domain o a ac ion XN(Ω)can
be enla ged by wo me hods: ei he inc easing he p edic-
ion ho izon N(since a g ea e p edic ion ho izon N1>N2
yields XN2(Ω)⊆XN1(Ω)) o conside ing a bigge e mi-
nal se (since Ω1⊆Ω2leads o XN(Ω1)⊆XN(Ω2)). The
i s way inc eases he numbe o decision a iables, and
hence, he compu a ional bu den o he op imiza ion p ob-
lem o be sol ed on-line, whils in he second he op imiza-
ion p oblem is simila . This second me hod is mo e con e-
nien and i has been used in se e al pape s such as (Magni
e al. 2001, Chen e al. 2001, Limon, Gomes da Sil a, Alamo
& Camacho 2003).
5 MPC based on a con ac i e e minal cons ain
Le us conside a sys em gi en by (1), subjec o cons ain s
on s a es (2) and on con ol ac ions (3). Unde he assump-
ion ha a sequence o N eachable se s {Ωi}is a ailable,
he ollowing op imiza ion p oblem is es ablished a sample
3
ins an k,
min
uVN(xk,u)
s. .x(i)∈X,u(i)∈U,i=0,···,N−1
x(N)∈Ωj,j=max(N −k,0)(5)
whe e x(i) = ϕ(i;xk,u). This p oblem is simila o he s an-
da d o mula ion, bu subs i u ing he e minal cons ain by
he con ac i e cons ain (5). The e minal se a ime 0 is
ΩN , and o he i s N sample imes, he index jin Ωjis
educed un il k=N , when he e minal se is Ω. The e o e,
he con ol law de i ed om his p oblem is ime- a ying
o he i s N sample imes. Fo k≥N he con ol law is
he same as ha o he ( ime-in a ian ) MPC wi h e minal
se Ω.
No e ha he op imiza ion p oblem can be sol ed on line
wi h simila compu a ional cos and ha he main compu a-
ion equi ed is he calcula ion o he con ac i e sequence
{Ωi}, which is done o line. In he ollowing heo em i is
p o ed ha he p oposed MPC con olle s abilizes he sys-
em asymp o ically in XN(ΩN ).
Theo em 4 Le a sys em gi en by (1) be subjec o con-
s ain on s a e (2) and on con ol ac ions (3). Le Ωbe
a posi i ely in a ian se o he sys em and le F(x)be an
associa ed Lyapuno unc ion such ha he assump ions o
heo em 3 a e sa is ied. Le {Ωi}be a sequence o N each-
able se s wi h Ω0=Ω. Then he sys em con olled by he
p oposed MPC is asymp o ically s able, wi h a domain o
a ac ion XN(ΩN ).
P oo : Fi s , he easibili y o he con olle is p o ed by in-
duc ion. Le xkand ukdeno e he s a e and he con ol ac ion
applied o he sys em a sampling ime k. Le us conside
ha he p oblem is easible a k=i, ha is, xi∈XN(ΩN −i);
hen he e is a sequence o Ncon ol ac ions which s ee s
he s a e o ΩN −i. Thus, gi en ha no misma ches exis be -
ween he nominal and he eal sys em, xi+1∈XN−1(ΩN −i).
Taking in o accoun lemma 2, i yields xi+1∈XN(ΩN −i−1).
Then, he op imiza ion p oblem is easible a k=i+1. Thus,
i x0∈XN(ΩN ) hen by induc ion i is in e ed ha he con-
olle is easible o all k<N . Since xN ∈XN(Ω), and be-
cause he e minal se is Ω o k≥N , hen he op imiza ion
p oblem will be easible all he ime in i ue o heo em 3.
The s abili y is de i ed om he ac ha he sys em e ol es
o XN(Ω)a e N samples. Fo k≥N , he op imiza ion
p oblem is he same as he s anda d MPC and, gi en ha he
assump ions o heo em 3 a e sa is ied, he sys em e ol es
asymp o ically o he o igin. 2
No e ha , i he assump ions p oposed in (Scokae , Mayne
& Rawlings. 1999) hold o k≥N , hen he op imali y o
he solu ion is no necessa y o gua an ee he asymp o ic
s abili y.
Rema k 5 (Enla gemen o he domain o a ac ion)
(i) I Ω⊂ΩN hen he p oposed con olle enla ges he
domain o a ac ion o he con olle , i.e. XN(Ω)⊆
XN(ΩN ).
(ii) I he se ΩN does no include Ω, hen he enla gemen
can be gua an eed by a simple p ocedu e: conside
any ini ial s a e x0∈XN(SN
i=0Ωi), hen a j such ha
x∈XN(Ωj)can be ound and he con ac ion can be
begun om i .
(iii) I he one-s ep se is compu ed accu a ely o ob aining
he sequence {Ωi}, hen XN(ΩN ) = XN+N (Ω). Hence,
he domain o a ac ion o he p oposed con olle is
he same as ha ob ained by s anda d MPC wi h p e-
dic ion ho izon N+N , bu conside ing only N con ol
ac ions as decision a iables.
Rema k 6 (Local op imali y) Since o k ≥N he op i-
miza ion p oblem o he p oposed con olle is he same as
ha o MPC wi h e minal egion Ω, i s solu ion is he same
and e ains he local op imali y o s anda d MPC. Fu he -
mo e, i has been p o ed ha unde he s abilizing condi-
ions o heo em 3, he e is a neighbo hood o he o igin
(which con ains he e minal egion Ω) whe e he e minal
cons ain is no longe ac i e and can be emo ed om he
op imiza ion p oblem (Limon, Alamo & Camacho 2003).
Consequen ly, in his egion he op imali y o he solu ion
depends on he chosen e minal cos , bu no on he (con-
ac i e) e minal egion.
Rema k 7 (Robus ness) Thanks o i s asymp o ic s abili y,
he p oposed MPC con olle e ains a ce ain deg ee o
obus ness o hose unce ain ies ha a e small enough,
as in he case o he s anda d o mula ion o MPC (Limon,
Alamo & Camacho 2002, Scokae , Rawlings & Meadows
1997). I a obus design o he MPC is ca ied ou , o
ins anceby means o a closed-loop o mula ion (Mayne e al.
2000), hen he p oposed idea can s ill be applied. The only
equi emen ha should be added is ha he sequence o
e minal se s be a sequence o obus ly eachable se s o he
obus in a ian e minal egion. Thus, he compu a ion o
he app oxima e one s ep se mus be obus ; ha is, o all
possible unce ain ies.
The p oposed MPC is ela ed o ha p esen ed in (Magni
e al. 2001), as bo h o hem enla ge he domain o a ac ion
o he MPC by conside ing a la ge e minal se . Howe e ,
bo h app oaches a e di e en , and in some way, complemen-
a y. In Magni’s MPC a p edic ion ho izon, Np, la ge han
he con ol ho izon, Nc, is conside ed and he local con ol
law is used o p edic he e olu ion om Nc o Np. This is
equi alen o conside ing a e minal cos gi en by
FNc,Np(x(Nc)) =
Np−1
∑
i=Nc
`(x(i),h(x(i)))+F(x(Np)) (6)
whe e x(i) = (x(i−1),h(x(i−1))) o i=Nc+1,···,Np
4
and he e minal egion gi en by ΩNp−Ncde i ed om (4)
using Qap(·) = Qh(·). The main no el y is ha his se is no
compu ed explici ly, bu implici ly desc ibed by he de ining
equa ions and added as e minal cons ain in he op imiza-
ion p oblem.
The MPC p oposed in his pape exploi s he no ion o con-
ol in a iance: he e minal se is eplaced by a sequence
o eachable se s compu ed o -line om (4) using an ap-
p oxima e ac able app oach Qap(·) o he one s ep se Q(·).
Since Qh(Ω)⊆Q(Ω) o any Ω, ou app oach can po en-
ially p o ide a la ge domain o a ac ion han Magni’s one
(as can be seen in he examples); his depends on how good
he app oxima ion Qap(·)is wi h ela ion o Qh(·). No e also
ha i he e minal cos (6) is conside ed, bo h app oaches
p o ide he same solu ion in a neighbo hood o he o igin.
I is wo h no ing ha he ex ension o he obus case o he
MPC p oposed in his pape is achie ed in a less in ol ed
way han Magni’s ex ension.
6 Examples
Example 1: Conside a second o de uns able linea sys em
gi en by x+=A·x+B·uwhe e
A="1.2775 −1.3499
1.0 0.0#B="1.0
0.0#
he cons ain s a e kxk∞≤5, |u|<1. The cos is gi en by
`(x,u) = kxk2
2+kuk2
2.
The sys em is con olled by an LQR con ol law and he
associa ed maximal posi i ely in a ian se is Ω(see Fig.1).
Based on Ω, he con ac i e sequence o N =5 con ol
in a ian se s has been calcula ed accu a ely, and hen Ωi=
Xi(Ω). The p edic ion and con ol ho izon is conside ed o
be N=3. In Fig.1 he domain o a ac ion o he p oposed
MPC, X3(Ω5), and he one o he o iginal MPC (e en wi h a
la ge p edic ion ho izon) X3(Ω)a e depic ed by a solid line.
In his case X3(Ω5) = X8(Ω), and he e o e he p oposed
con olle is able o s abilize wi h N=3 anys a e s abilizable
by he o iginal MPC wi h N=8. In his igu e he ajec o ies
o he s a es o he sys em a e plo ed. As can be seen, he
s a e e ol es asymp o ically o he o igin.
Example 2: Conside he sys em used in (Chen & Allg¨
owe
1998) desc ibed by
˙x1=x2+u·(µ+(1−µ)·x1)
˙x2=x1+u·(µ−4·(1−µ)·x2)
whe e he pa ame e µis 0.5. The inpu is cons ained o
|u| ≤ 2. The sys em has been disc e ized using a 4 h o de
Runge-Ku a me hod wi h a sampling ime o 0.1 ime-uni s.
The s age cos is gi en by `(x,u) = 0.5kxk2
2+kuk2
2.
−4 −3 −2 −1 0 1 2 3 4
−4
−3
−2
−1
0
1
2
3
4
x1
x2
X3(Ω5)
X3(Ω)
Ω
Fig. 1. E olu ion o he sys em o example 1
The sys em is locally asymp o ically s abilized by a local lin-
ea con olle u=h(x)wi h an associa ed Lyapuno unc ion
F(x) = 16.5926(x2
1+x2
2)+23.1852x1x2in he posi i ely in-
a ian se Ω={x∈IR2:F(x)≤0.7}. Bo h o hem sa is y
he assump ions o heo em 3. A sequence o 10 eachable
se s has been compu ed o line using as app oxima ion o
he one-s ep se he one p oposed in (B a o e al. 2003).
Based on his sequence, he p oposed MPC echnique has
been applied o he sys em wi h a con ol ho izon o Nc=3.
The conside ed e minal cos is gi en by (6) conside ing a
p edic ion ho izon o 33. The sequence o se s and he closed
loop s a e po ai a e shown in igu e 2.
−1 0 1
−4
−3
−2
−1
0
1
x1
x2
A
B
C
D
E
F
Fig. 2. The sequence o eachable se s and s a e po ai o he
sys em o example 2
I is wo h ema king ha none o he depic ed ini ial s a es
a e easible o a s anda d MPC wi h p edic ion and con ol
ho izon o 3. I Magni’s MPC is used wi h Np=33 and
Nc=3, hen he ini ial s a es A,B,E and F a e easible, while
C and D a e only easible o he p oposed MPC.
7 Conclusions
In his pape a o mula ion o MPC o enla ge he domain o
a ac ion wi hou inc easing he p edic ion ho izon is p e-
sen ed. I is based on subs i u ing he s anda d in a ian e -
minal egion by a sequence o eachable se s, and hence, he
e minal cons ain by a con ac i e e minal cons ain . This
sequence o se s can be compu ed by a p oposed me hod
5

based on he calcula ion o an inne app oxima ion o he
one-s ep se . The p oposed con olle s abilizes he sys em
unde he same assump ions as he MPC wi h e minal con-
s ain , gua an eeing he enla gemen o he domain o a -
ac ion as well as he local op imali y. I is also shown ha
his idea can be s aigh o wa dly ansla ed o he obus
case.
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6