IMPROVED MPC DESIGN BASED ON SATURATING CONTROL
LAWS
D. Limon∗, J.M. Gomes da Sil a J .†, T. Alamo∗and E.F. Camacho∗
∗Dp o. de Ingenie ia de Sis emas y Au om´
a ica. Uni e sidad de Se illa
Camino de los Descub imien os s/n, 41092 Se illa, Spain
e-mail:{limon,alamo,edua do}@ca uja.us.es
†UFRGS - Dep o. de Engenha ia El´
e ica
A . Os aldo A anha 103, 90035-190 Po o Aleg e-RS, B azil
e-mail:[email p o ec ed]
Keywo ds: model p edic i e con ol, cons ained con ol,
con ol sa u a ion, in a ian se s, asymp o ic s abili y.
Abs ac
This pape is conce ned wi h he design o s abilizing MPC
con olle s o cons ained linea sys ems. This is achie ed by
ob aining a sui able e minal cos and e minal cons ain using
a sa u a ing con ol law as local con olle . The sys em con-
olled by he sa u a ing con ol law is modeled by a poly opic
di e en ial inclusion. Based on his, i is shown how o de e -
mine a Lyapuno unc ion and a polyhed al in a ian se which
can be used as e minal cos and cons ain . The ob ained in-
a ian se is po en ially la ge han he maximal in a ian se
o he unsa u a ed linea con olle , O∞.
Fu he mo e, conside ing hese elemen s, a dual-mode MPC is
p oposed. This con olle gua an ees he enla gemen o he
domain o a ac ion o , equi alen ly, he educ ion o he p e-
dic ion ho izon o a gi en se o s abilizable ini ial s a es. I
he local con ol law is he sa u a ing LQR con olle , hen he
p oposed dual-mode MPC con olle main ains he local op i-
mali y o he s anda d MPC. An illus a i e example is gi en.
1 In oduc ion
MPC has become a popula con ol echnique bo h in academy
and indus y. The main eason o his success is ha MPC can
be conside ed as an op imum con ol echnique able o deal
wi h cons ain s on he s a es and he manipula ed a iables in
an explici manne . Fu he mo e, a heo e ical amewo k o
analyze opics such as s abili y, obus ness, op imali y, e c. has
been de eloped ecen ly. See [9] o a su ey, o [2] o p ocess
indus y applica ion issues.
I has been p o ed [9] ha closed-loop s abili y o he MPC
con olle is gua an eed by adding a e minal cos and a e -
minal cons ain in he op imiza ion p oblem. The conside ed
e minal cos is a Lyapuno unc ion associa ed o he sys em
con olled by a local con ol law. An associa ed in a ian se
is he e minal se . The domain o a ac ion o he MPC con-
olle is he se o s a es ha can be s ee ed o he e minal
egion in Ns eps, whe e Nis he p edic ion ho izon.
The mos common way o designing a s abilizing MPC con-
olle o a cons ained linea sys em is based on he ollowing
h ee ing edien s: (i) an LQR as local s abilizing con olle ; (ii)
a quad a ic e minal cos equal o he op imal cos ; (iii) a e mi-
nal se equal o he maximal admissible in a ian se , O∞.In
his case, he ob ained MPC con olle is equal o he in ini e-
ho izon op imal con olle (cons ained LQR) in a neighbo -
hood o he o igin.
I a sa u a ing con ol law is used ins ead, hen he egion whe e
he local con olle is s abilizing can be inc eased. The e o e,
he domain o a ac ion o he MPC con olle can be enla ged
(o , equi alen ly, he equi ed p edic ion ho izon can be e-
duced o a gi en se o s abilizable ini ial s a es). This idea
has been p e iously used in [3] o single-inpu linea sys ems
subjec o cons ain in he inpu (bu no in he s a es). A e-
gion whe e he sa u a ed LQR is op imal is p esen ed and he
op imal cos is explici ly compu ed o he closed-loop sys em.
This cos is alid in he maximal in a ian se con ained in ha
egion. This in a ian se may be non-con ex, which leads o a
non-con ex MPC op imiza ion p oblem.
In his pape we use a sa u a ing con ol law om a di e en
poin o iew. F om a di e en ial inclusion ep esen a ion, i
is shown how o compu e a sui able Lyapuno quad a ic unc-
ion and a polyhed al in a ian se o he cons ained sys em
con olled by he sa u a ing con ol law. The ob ained in a i-
an se is po en ially la ge han O∞. Hence, he domain o
a ac ion o he ob ained s abilizing MPC con olle may be
enla ged. Fu he mo e, he ob ained MPC is based on a con ex
quad a ic p og amming p oblem and can be applied o gene ic
cons ained mul i-inpu sys ems. In o de o gua an ee he en-
la gemen o he domain o a ac ion emaining he local op-
imali y o he LQR-based design, a dual MPC echnique is
p oposed.
No a ions. Fo any ec o x∈IR n,x0means ha all
he componen s o x, deno ed x(i), a e nonnega i e and x0
means ha a e s ic ly posi i e. y=|x|, o x∈IR n, deno es
he componen -wise absolu e alue, ha is, y(i)=|x(i)|.Fo
wo ec o s x,yo IR n, he no a ion xymeans ha x−y
0. Fo a symme ic ma ix A,A>0means ha i is de ini e
posi i e. Consequen ly, o wo symme ic ma ices, Aand B,
A>Bmeans ha A−B>0. Fo a de ini e posi i e ma ix
P>0,ε(P,α)deno es he ellipsoid ε(P,α)={x∈IR n:
xTPx ≤α}.A(i)deno es he i h ow o ma ix A, and AT
deno es he anspose o A.Indeno es he n-o de iden i y
ma ix. Fo any ec o x∈IR n,diag(x)deno es he diagonal
ma ix ob ained om x.Co {·} deno es a con ex hull.
2 Sa u a ing Con ol Laws
Le a disc e e- ime linea sys em be desc ibed by:
x+=Ax +Bu (1)
whe e x∈IR nis he cu en s a e o he sys em, u∈IR mis
he cu en inpu and x+is he successo s a e. The sys em is
subjec o ha d cons ain s on s a e and con ol inpu :
x(k)∈X, u(k)∈U
o all k≥0. The se s Xand Ua e polyhed a con aining he
o igin in hei in e io . Fu he mo e he se Uis gi en by
U={u∈IR m:|u|ρ}
whe e he ec o ρ∈IR mis such ha ρ0.
Conside ha sys em (1) is s abilized by a s a e eedback law
u=Kx, ha is, Kis such ha he eigen alues o (A+BK)
a e placed inside he uni disk. No e ha his con ol law is
admissible inside a polyhed al egion de ined as:
RL={x∈IR n;|Kx|ρ}(2)
and hence, he con ol law u=Kx is only able o s abilize he
cons ained sys em in a subse o RL∩X. This se is he max-
imal posi i ely in a ian se o x+=(A+BK)xcon ained
in RL∩X(also called maximal admissible se ) and deno ed as
O∞[4].
The con olle migh be ex ended ou side his egion, conside -
ing ha he e ec i e con ol law o be applied o he sys em is a
sa u a ed s a e eedback, whe e each componen o he con ol
ec o is de ined as ollows:
u(i)=


−ρ(i)i K(i)x<−ρ(i)
K(i)xi −ρi≤K(i)x≤ρ(i)
ρ(i)i K(i)x>ρ
(i)
(3)
Fo all i=1,···,m. In his case, he closed-loop sys em be-
comes
x+=Ax +Bsa (Kx)(4)
which is a non-linea sys em.
No e ha each componen o he con ol law de ined by (3) can
also be w i en as [6, 5]:
u(i)=sa (K(i)x)=α(x)(i)K(i)x(5)
whe e
α(x)(i)

=




−ρ(i)
K(i)xi K(i)x<−ρ(i)
1i −ρ(i)≤K(i)x≤ρ(i)
ρ(i)
K(i)xi K(i)x>ρ
(i)
(6)
wi h 0<α(x)(i)≤1,i=1,...,m.
The coe icien α(x)(i)can be iewed as an indica o o he de-
g ee o sa u a ion o he i h en y o he con ol ec o . In ac ,
smalle is α(x)(i), a he is he s a e ec o om he egion RL
gi en by Equa ion (2). No ice ha α(x)(i)is a unc ion o he
cu en s a e x. Fo he sake o simplici y, in he sequel we de-
no e α(x)(i)as α(i). Hence, de ining bo h α∈IR mas a ec o
o which he i h en y is α(i),i=1,...,m, and a diagonal
ma ix D(α)
=diag(α), he sys em (4) can be e-w i en as
x+=(A+BD(α)K)x(7)
No e ha he ma ix (A+BD(α)K)depends on he cu en
s a e x, since αdoes.
Conside now a ec o α∈IR msuch ha α(i)∈(0,1] and
de ine he ollowing polyhed al egion
RL(α)={x∈IR n:|Kx|ρ(α)}(8)
whe e ρ(α)(i)=ρ(i)
α(i),i=1,...,m. Hence, o all x∈
RL(α), i ollows ha α(i)≤α(i)≤1,∀i=1,...,m. No e
ha RL⊆RL(α).
F om con exi y a gumen s, o all xbelonging o RL(α), i ol-
lows ha : D(α)∈Co{D1(α),D
2(α),...,D
2m(α)}whe e
{Dj(α)}a e diagonal ma ices whose diagonal elemen s can
assume he alue 1o α(i). The e o e, he sys em (4) can be
locally ep esen ed by a poly opic model (o a poly opic di e -
en ial inclusion) as i is s a ed in he ollowing lemma [5, 6].
Lemma 1 Conside sys em (4) and a ec o α∈IR mwhose
componen s α(i),i =1,...,mbelong o he in e al (0,1].I
x∈RL(α), hen successo s a e x+de i ed om he sys em
(4) can be compu ed by he ollowing poly opic model:
x+=
2m

j=1
λjAj(α)x(9)
wi h 2m

j=1
λj=1,λj≥0and whe e
Aj(α)
=A+BDj(α)K
No e ha he ac o s λjmay depend on he cu en s a e x.
F om his esul he nex lemma can be de i ed.
Lemma 2 I a se S∈IR ncon ained in he egion RL(α)
is posi i ely in a ian o he poly opic sys em (9), hen Sis
posi i ely in a ian o he sa u a ed sys em (4).
Based on he poly opic di e en ial inclusion ep esen a ion (9),
in he ollowing lemma a e gi en su icien condi ions o ind a
quad a ic Lyapuno unc ion o he sa u a ed sys em (4). This
lemma is pa icula ly use ul o he design o he s abilizing
MPC con olle p oposed in sec ion 5.
Lemma 3 Conside sys em (4), a ec o α∈IR mwi h α(i)∈
(0,1], and posi i e de ini e ma ices R∈IR m×mand Q∈
IR n×n. I he e exis s a posi i e de ini e ma ix P∈IR n×n
sa is ying he ollowing LMI
Aj(α)TPAj(α)−P
+KTDj(α)TRDj(α)K+Q<0(10)
o all j=1,...,2m, hen he unc ion F(x)=xTPx e i ies
F(x+)−F(x)≤−xTQx (11)
−sa (Kx)TRsa (Kx)
o all x∈RL(α), whe e x+=Ax +Bsa (Kx).
P oo : I ollows di ec ly om he applica ion o Schu ’s com-
plemen , con exi y a gumen s and om ep esen a ion o he
sa u a ed sys em gi en by Equa ion (9).
3 De e mina ion o he Te minal Se
In his sec ion, we show how o compu e a sui able posi i ely
in a ian se o sys em (4), based on he poly opic ep esen a-
ion p esen ed in he p e ious sec ion. The p ocedu e p o ides
he maximal in a ian se con ained in XL=RL(α)∩X o
he poly opic sys em (9).
Conside he ollowing sequence o admissible se s o he sys-
em (9) , gi en by
C0=XL
Ck=Q(Ck−1)∩XL,k≥1(12)
whe e he se Q(Ω) is he one-s ep se o Ω, ha is, he se o
s a es ha each Ωin one s ep [1, 7].
The se Ckis he egion o ini ial s a es om which he sys em
e olu ion emains in XL o he nex ksampling imes. The
sequence o admissible se s sa is ies ha Ck+1 ⊆Ck. The se
C∞is he se o s a es ha a e kep in XL o all he ime, and
hence is he maximum posi i ely in a ian se con ained in XL
o he poly opic sys em.
The one-s ep se o a poly ope Ω={x∈IR n:Hx h} o
he poly opic sys em (9) is ano he poly ope gi en by
Q(Ω) =
2m

j=1
Qj(Ω)
whe e Qj(Ω) is he one-s ep se o Ω o he sys em x+=
Aj(α)x, ha is, Qj(Ω) = {x∈IR n:HAj(α)xh}.
Hence, p o ided ha XLis a poly ope, i ollows ha Ckis
a poly ope, since i is he in e sec ion o se e al poly opes.
In he nex heo em i is s a ed ha he maximum in a ian se
C∞is ini ely de e mined and i is a compac poly ope, bu i s
he ollowing lemma is p esen ed:
Lemma 4 I he e exis s a ma ix o he poly opic model Aj(α)
such ha he pai (K, Aj(α)) is obse able, hen he se Cn−1
is a compac poly ope, whe e nis he o de o he sys em.
P oo :
Le Cj
n−1be he admissible se in n−1s eps o he sys em
x+=Aj(α)xin he se XL(α), i.e. Cj
n−1={x∈IR n:
Aj(α)ix∈XL,i =0,···,n −1}. Then i is clea ha
Cn−1⊆Cj
n−1. Fu he mo e
Cj
n−1⊆{x∈IR n:|KAj(α)ix|ρ(α),i=0,···,n−1}
Taking in o accoun ha he obse abili y ma ix o (K, Aj(α))
is ull- ank, hen Cj
n−1is compac and, hence, Cn−1is com-
pac .
Theo em 1 Le V(x)=xTPxbe a Lyapuno unc ion o he
poly opic sys em (9) such ha o all x∈RL(α),V(x+)≤
µV (x)whe e µ∈(0,1) . I he e exis s a ma ix Aj(α)such
ha (K, Aj(α)) is obse able hen:
(i) C∞is ini ely de e mined
(ii) C∞is a posi i ely in a ian se o he sa u a ed sys em
(4), whe e i is exponen ially s able and sa is ies he con-
s ain s.
P oo :
F om lemma 4 i is de i ed ha he se Cn−1is compac , and
hence Ckis compac o all k≥n−1, since Ck⊆Cn−1.
Le ε(P,β)deno e he ellipsoid {x∈IR n:xTPx ≤β}.
Since Cn−1is bounded, he e is a ini e βsuch ha Cn−1⊂
ε(P,β).
Le ε(P,γ)be he maximum ellipsoid such ha ε(P,γ)⊂XL.
Since xTPx is a Lyapuno unc ion s ic ly dec easing o all
x∈XL, his se is a ( µ-con ac i e )posi i ely in a ian se o
he poly opic sys em and, hence, ε(P, γ)⊂C∞.
No e ha ε(P,γ)⊂C∞⊆Cn−1⊂ε(P,β)and hence β≥γ.
Le Mbe a cons an such ha βµM≤γ, hen, p o ided ha
V(x+)≤µV (x), i ollows ha o all x∈ε(P,β) he s a e
o he sys em eaches ε(P,γ)in Ms eps o less.
Conside i≥n−1+M, hen Ci⊆Cn−1⊂ε(P,β). Conse-
quen ly, since Ci⊆CM, o all x∈Ci he sys em e olu ion is
con ained in XLand eaches ε(P,γ)⊂C∞in Ms eps o less.
The e o e, o all x∈Ci, we ha e ha he sys em emains
in XL o all he ime and hence Ci⊆C∞. This yields o
C∞⊆Ci⊆C∞which p o es ha Ci=C∞and, he e o e,
i is ini ely de e mined.
C∞is he maximal in a ian se con ained in XL o he poly-
opic sys em (9) and hen, since XL⊆RL(α), i is also a
posi i ely in a ian o he sa u a ed sys em (4). Fu he mo e,
he exis ence o a s ic ly dec easing Lyapuno unc ion V(x)
o he poly opic sys em, ensu es he exponen ial s abili y o
(4) in he se C∞.
No ice ha he obse abili y condi ion on (K, Aj(α)) is no
necessa y i he se XLis compac . This can be gua an eed i
Xo RL(α)a e compac .
The ob ained posi i ely in a ian se C∞is a poly ope, bu i
is no possible o ensu e ha he maximum in a ian se o he
unsa u a ed con ol law O∞is always con ained in C∞.How-
e e , as could be seen in he nume ical example, his inclusion
o en occu s o , a leas , C∞is po en ially la ge ha O∞.
No e ha C∞is con ained in RL(α)and in lemma 3, αis sup-
posed o be gi en. O cou se, in o de o ob ain a la ge egion
RL(α)and, as a consequence, a la ge in a ian se C∞,i is
in e es ing o e i y (10) wi h αha ing componen s as small
as possible. In he 1-inpu o 2-inpu cases, by applying a g id
sea ch, one can easily de e mine he minimal α o which i
is possible o ind a solu ion o (10). Conside ing he gene ic
mul i-inpu sys ems, i e a i e schemes, as p oposed in [6, 5],
can be used.
Fo he compu a ion o he admissible se s using he p oposed
p ocedu e, an algo i hm o emo ing edundan inequali ies
and ano he one o subse es ing o polyhed a a e necessa y.
The e exis s e icien algo i hms o hese asks [7]. A di e en
algo i hm o he compu a ion o C∞based on linea p og am-
ming schemes is gi en in [5].
4 MPC S abili y
In he p e ious sec ions, a p ocedu e o compu e a Lyapuno
unc ion and an associa ed in a ian se o he sys em con-
olled by a sa u a ing con ol law has been p oposed. These
ing edien s can be used o design a s abilizing MPC con olle .
In MPC, he con ol ac ion o a gi en s a e xis ob ained by
sol ing an op imiza ion p oblem PN(x)de ined by
V0
N(x) = min
u
VN(x, u)
s. .
u(j)∈U, x(j)∈X, j =0,···,N −1
x(N)∈X
whe e u={u(0),u(1),···,u(N−1)}is a sequence o N
con ol ac ions, VN(x, u)is gi en by
VN(x, u)=
N−1

j=0
L(x(j),u(j)) + F(x(N))
whe e L(x, u)=x2
Q+u2
R, wi h Q>0and R>0and
x(j)=xu(j, x), ha is he s a e a ime ji he ini ial s a e is
xa ime 0and he con ol sequence uis applied o he sys em.
A e en (k,x)p oblem PN(x)is sol ed, yielding he mini-
mize u0and he op imal cos V0
N(x). The MPC con ol law is
implici ly gi en by u=κN(x)=u0(0).
In [9] he well-known su icien condi ions o gua an ee asymp-
o ic s abili y o he MPC con olle a e s a ed.
Theo em 2 [9] I he e minal se X is a posi i ely in a ian
se o he sys em con olled by a local con ol law u=κ (x)
such ha κ (x)∈U o all x∈X
and he e minal cos F(x)
is an associa ed Lyapuno unc ion such ha
F(Ax +Bκ (x)) −F(x)≤−L(x, κ (x)) ∀x∈X
hen u=κN(x)asymp o ically s abilizes he sys em o all
easible ini ial s a e, i.e. x0∈XN(X ).
The domain o a ac ion XN(X )may be enla ged by inc eas-
ing he p edic ion ho izon (which yields a g ea e compu a-
ional bu den) o inc easing he size o he e minal se [8].
I is wo h ema king ha i one chooses he LQR as local
con olle κ (x)=KLQRx, he uncons ained op imal cos
F(x)=xTPLQRxas e minal cos and he maximal in a ian
se O∞as e minal se , he ob ained MPC con ol law is he
in ini e ho izon op imal con ol law in a neighbo hood o he
o igin.
5 MPC con olle design
Based on he p e ious p esen ed esul s, he ollowing heo em
can be s a ed.
Theo em 3 Conside a locally s abilizing con olle u=
sa (Kx)and le Pbe he ma ix solu ion o he equa ion (10)
o gi en weigh ing ma ices Qand Rand o a gi en ec o
α. Suppose ha C∞de ined om he sequence (12) is ini ely
de e mined. Then he MPC con olle ob ained by conside ing
F(x)=xTPxas e minal cos and X =C∞as e minal se
s abilizes asymp o ically he sys em o all s a e in XN(C∞).
No e ha his choice makes he domain o a ac ion o he
MPC po en ially la ge ha he one based on he uncons ained
local con olle and he esul an op imiza ion p oblem is a con-
ex quad a ic p og amming one.
Al hough a sa u a ed LQR, u=sa (KLQRx), is used as local
con olle , he designed MPC may no be he in ini e-ho izon
op imal in a neighbo hood o he o igin. This is due o he ac
ha he conside ed e minal cos is a conse a i e app oach o
he op imal cos o he con olle and hence, P>P
LQR.In
o de o educe his conse a i eness, he ma ix Pis compu ed
by sol ing (10) minimizing i s ace. Fu he mo e, he ob ained
e minal se C∞may no include O∞.
These d awbacks can be o e come by a simple p ocedu e o
implemen he con olle :
•I x∈XN(O∞), hen conside F(x)=xTPLQR xand
X =O∞.
•Else, conside F(x)=xTPxand X =C∞.
The con ol law is ob ained by sol ing he esul an op imiza-
ion p oblem which yields a dual-mode con ol law. This
con olle asymp o ically s abilizes he sys em in a domain o
a ac ion XN(C∞)∪XN(O∞)and hence, he enla gemen
p ope y is ensu ed. Mo eo e , he in ini e ho izon op imali y
p ope y o he MPC holds.
The condi ion x∈XN(O∞)can be easily checked, since his
se is a polyhed al ha can be compu ed e icien ly o -line [7].
Ano he echnique is checking he easibili y o he associa ed
op imiza ion p oblem o a gi en x, which can be posed as an
LP p oblem.
As i was men ioned be o e, he MPC design based on a sa u-
a ed LQR p oposed in [3] main ains he op imali y p ope y
o he MPC a expense o using a non-con ex e minal se
(and hence a non-con ex op imiza ion p oblem). This non-
con exi y can be o e come by choosing a la ge enough p e-
dic ion ho izon. This p ocedu e inc eases he compu a ional
bu den and may educe he enla gemen o he domain o a -
ac ion de i ed om he p oposed design. Fu he mo e, his
echnique is only alid o single-inpu sys ems.
The design p esen ed in his pape p o ides a poly ope and a
quad a ic e minal cos o mul iple-inpu sys ems subjec o
cons ain s on s a es. This yields a s abilizing MPC de i ed
om a con ex op imiza ion p oblem. Fu he mo e, by using a
simple dual-mode MPC con olle , he op imali y and he en-
la gemen p ope ies a e gua an eed.
6 Nume ical Example
Conside a sys em x+=Ax +Bu gi en by
A=11
01
B=00.5
10.5
whe e he inpu s a e cons ained o u∞≤0.3and x∞≤
2. Fo his sys em, a LQR con olle wi h Q=I2and R=I2
is compu ed. The con olle u=Kx and he op imal cos
F(x)=xTPLQRxa e gi en by
K=−0.0037 −0.5850
−0.5919 −0.8844 PLQR =2.1801 1.1838
1.1838 2.7688 
The maximal in a ian se o his con olle , O∞, is shown in
igu e 1.
Following he echnique p oposed in he pape , a Lyapuno
unc ion and a posi i ely in a ian se o he sa u a ing con-
ol law de i ed om he LQR con olle is compu ed. Fi s ,
i has been chosen a ec o αsuch ha he LMI (10) is easi-
ble. The ob ained ec o is α=[0.25,0.2] and he calcula ed
quad a ic e minal cos F(x)=xTPxis gi en by
P=33.5508 28.2391
28.2391 208.3942 
Nex , he maximal in a ian se , C∞, o he poly opic sys em
con ained in XL=RL(α)∩Xis calcula ed. Bo h se s a e
depic ed in igu e 1. No e ha O∞⊂C∞.
−2 −1 0 1 2
−2
−1
0
1
2
x1
x2
O∞
C∞
XL
Figu e 1: Te minal se s O∞and C∞.
Conside ing he e minal cos and e minal se ob ained, a s a-
bilizing MPC can be compu ed. The domain o a ac ion o
he MPC based on he unsa u a ed LQR con olle , XN(O∞),
is con ained in he one o he MPC based on he sa u a ing con-
ol law, XN(C∞). This enla gemen is shown in igu e 2 o
an MPC wi h N=2.
−2 −1 0 1 2
−2
−1
0
1
2
x1
x2
X2 ( C∞ )
X2 ( O∞ )
Figu e 2: Domain o a ac ion o he MPC con olle wi h each
e minal se .
No e ha he enla gemen o he domain o a ac ion is equi -
alen o a educ ion o he p edic ion ho izon. In igu e 3, i is
shown ha X4(O∞)⊂X2(C∞). The e o e, all s a e s abiliz-
able by he LQR based MPC wi h N=4is s abilizable by he
MPC based on he sa u a ed LQR wi h N=2. Fu he mo e,
in his case, X4(C∞)is equal o he maximal s abilizable se
X∞. Hence, o N=4, he MPC based on he sa u a ing con-

−2 −1 0 1 2
−2
−1
0
1
2X4 (C∞ ) = X∞X2 (C∞ )
X4 (O∞ )
O∞
C∞
x1
x
2
Figu e 3: Compa ison be ween domains o a ac ion o he
MPC o se e al p edic ion ho izons.
ol law is able o s abilize all s abilizable se , while he MPC
based on he LQR is no .
Howe e , al hough he LQR based MPC is he op imal con-
olle , he MPC based on he sa u a ing LQR is no . In o de
o imp o e he op imali y o he con olle , a dual MPC is p o-
posed. In igu e 4, i is compa ed he e olu ion be ween he
dual MPC and he s anda d MPC based on he sa u a ing con-
ol law o ou ini ial s a es. In Table 1 he cos associa ed
o he e olu ion o he closed loop sys em o bo h con olle s
is compa ed. I is demons a ed ha he dual-mode MPC con-
olle p esen s a lowe cos , and hence, he pe o mance o he
closed loop sys em is be e .
−2 −1 0 1 2
−2
−1
0
1
2
4
1
3
2
dual MPC
s anda d MPC
X2 (C∞)
X2 (O∞)
Figu e 4: Closed loop s a e po ai o he dual MPC and he
s anda d MPC.
x0dual s anda d
1 13.2857 13.3125
211.3655 11.3759
312.7831 12.8275
414.6420 14.8448
Table 1: Compa a i e o he cos o he e olu ion o s anda d
MPC and he p oposed dual-mode MPC
7 Conclusions
In his pape we p esen a echnique o design a s abilizing
MPC con olle o cons ained linea sys ems, which is based
on a sa u a ing con ol law. Using a polyhed al di e en ial in-
clusion o ep esen ing he beha iou o he closed-loop sys-
em, a quad a ic e minal cos and a poly opic in a ian se can
be e icien ly compu ed. This se is po en ially la ge han he
maximal in a ian se o he sys em con olled by he unsa u-
a ed con olle . These ones can be used o design a s abilizing
MPC con olle wi h an associa ed con ex op imiza ion p ob-
lem. Fu he mo e, a dual-mode MPC con olle is p esen ed.
This app oach gua an ees he enla gemen o he domain o a -
ac ion and keeps he local op imali y p ope y de i ed om
using a LQR as local con olle .
Re e ences
[1] F. Blanchini. Ul ima e boundedness con ol o disc e e-
ime unce ain sys em ia se -induced lyapuno unc ions.
IEEE T ansac ions on Au oma ic Con ol, 39:428–433,
1994.
[2] E. F. Camacho and C. Bo dons. Model P edic i e Con ol.
Sp inge -Ve lag, 2 edi ion, 1999.
[3] J. A. De Don´
a, M. M. Se on, D. Q. Mayne, and G. C.
Goodwin. Enla ged e minal se s gua an eeing s abili y
o eceding ho izon con ol. Sys ems & Con ol Le e s,
47:57–63, 2002.
[4] E. G. Gilbe and K. Tan. Linea sys ems wi h s a e and
con ol cons ain s: The heo y and applica ion o maximal
ou pu admissible se s. IEEE T ansac ions on Au oma ic
Con ol, 36:1008–1020, 1991.
[5] J. M. Gomes Da Sil a J . and S. Ta bou iech. S abili y
egions o linea sys ems wi h sa u a ing con ols. In P o-
ceedings o he ECC, 1999.
[6] J. M. Gomes Da Sil a J . and S. Ta bou iech. Local s a-
biliza ion o disc e e- ime linea sys ems wi h sa u a ing
con ols: an LMI-based app oach. IEEE T ansac ions on
Au oma ic Con ol, 46:119–125, 2001.
[7] E. C. Ke igan. Robus Cons ain Sa is ac ion: In a i-
an Se s and P edic i e Con ol. PhD hesis, Uni e si y o
Camb idge, 2000.
[8] D. Limon, T. Alamo, and E. F. Camacho. Enla ging he
domaino a ac iono MPCcon olle usingin a ian se s.
In P oceedings o he IFAC Wo ld Cong ess, 2002.
[9] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M.
Scokae . Cons ained model p edic i e con ol: S abili y
and op imali y. Au oma ica, 36:789–814, 2000.