Improved MPC Design based on Saturating Control Laws

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. 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