On the design of Robust tube-based MPC for tracking

On hedesigno Robus ube-basedMPC o acking⋆ D.Limon∗I.Al a ado∗T.Alamo∗E.F. Camacho∗ ∗Dp o.deIngenie ´ıa deSis emasyAu om´a ica,Uni e sidad de Se illa,A da delosDescub imein os s/n41092,Se illa(limon, al a ado,alamo,camacho@ca uja.us.es). Abs ac : Thispape dealswi h he designp ocedu eo he ecen lyp esen ed obus MPC o acking o cons ained linea sys emswi haddi i edis u bances.Thiscon olle isbased onnominalp edic ions and i iscapable os ee he nominalp edic ed ajec o y o any a ge admissibles eady s a e, ha is e aining easibili yunde anyse poin change. Bymeans o he no iono ubeo ajec o ies, obus s abili yand con e gence isachie ed. The con olle o mula ionhas somepa ame e swhichp o ides ex adeg ees o eedom o he designp ocedu eo he p edic i econ olle .Theseallow odealwi hcon olobjec i es such asdis u bance ejec ion, ou pu o se p io i iza iono enla gemen o he domaino a ac ion. In hispape ,ou pu p io i iza ionme hod, LMIbased designp ocedu es and algo i hms o he calcula iono in a ian se sa ep esen ed. The p oposed enhanced designo he MPCis demons a ed byanillus a i eexample. 1.INTRODUCTION Model p edic i econ ol(MPC)isacon ol echnique capable odealwi hha dcons ain so he sys em and he op imiza iono ape o mance index.Thisisachie ed byposing he con olp oblem asama hema icalp o- g amming p oblem and applying he op imalsolu ionina eceding ho izonmanne .Thiscon olle ypically equi es a e minals a epenaliza ionand cons ain , ino de o ensu eclosed loops abili y.The s abilizing ing edien s and he ini ep edic ionho izonmake ha he p edic i e con olle mayloose easibili yunde se poin changes. Recen ly,ano el Robus MPC o acking hasbeen p e- sen ed (Al a ado e al. [2007]).The maincha ac e is ics o hiscon olle a e: (i)adds ana i icials eady s a e asadecision a iable(a i icials eady s a e) (ii)uses an in a ian se o acking asa e minalcons ain (iii) conside sacos unc ion penalizing he e o w. . he a i icials eady s a eand anaddi ional e mpenalizing he de ia ion be ween he a i icials eady s a eand he a ge s eady s a e(i )i isbased on he ube-based MPC con olle (Mayne e al. [2005]).Thiscon olle ,unde mildassump ions,cans ee he unce ainsys em inan admissiblee olu ion o a neighbo hoodo anyadmissible s eady s a e. The obus MPC o acking has somepa ame e s o be uned, whichp o ides ex adeg ees o eedom obe exploi ed acco ding o he con olobjec i es.Thisallows us odealwi hdis u bance ejec ion, enla gemen o he domaino a ac ionand ou pu acking p io i iza ion when he se poin isno admissible. In hispape , he e ec o eachpa ame e o he MPCis analysed and p ocedu es o hei selec iona ep esen ed. ⋆Theau ho sacknowledgeMCYT-Spain o unding hiswo k (con ac sDPI2007-66718-C04-01 and DPI2005-04568) Fi s , i is shownhow odesign he p oposed MPC o ensu e he e olu ion o anop imized s eady s a ewhen he a ge isno admissible. Then he obus localcon olle isdesigned ominimize he e ec o he dis u bances on he s a ee olu ionand hence oenla gede domaino a ac iono he MPC.Thiscan beposed asase o linea ma ixinequali ies whichsolu ionp o ides he con ol gain. Finally,exis ing esul son he compu a iono an app oxima ion o he minimal obus posi i elyin a ian se a especialized o dis u bances ha can be ep esen ed asazono ope(ana ine mapping o auni a ybox). The pape is s uc u ed as ollows:In§2 he p oblem o sol eisdesc ibed and in§3 he obus MPC o acking isp esen ed, in§4 he designp ocedu eisin oduced, in §5 he ool ha minimize he in a ian se .The pape inishes wi hanillus a i eexampleinsec ion5 and some conclusions. No a ion:Ade ini eposi i ema ixTisdeno ed as T>0 and T>Pdeno es ha T−P>0.Fo a gi en symme icma ixP>0,kxkPdeno es he weigh ed Euclideanno mo x, i.e. kxkP=√x⊤Px. (a,b)∆ =[a⊤,b⊤]⊤.Conside a∈IRna,b∈IRnb,and se Γ⊂IRna+nb, hen p ojec ionope a ionisde ined as P oja(Γ)={a∈IRna:∃b∈IRnb,(a,b)∈Γ}.Gi en wose sUand V,such ha U⊂IRnand V⊂IRn, he Minkowskisum isde ined byU⊕V∆ ={u+ |u∈U, ∈V}, he Pon yaginse di e ence is:U⊖V∆ ={u|u⊕V⊆U}. Fo a gi en ma ixM∈IRn×mand ase V⊂IRm, he se MV⊂IRndeno es he se {y=M , ∈V}.Fo a gi en λ,λX={λx:x∈X}.Le bea gene ic ec o de ined as ∆ ={ (0), (1),...}.Ase isaCse i iscompac ,con ex no emp y.Ama ix0n,m∈IRn×mdeno es ama ixo ze os.The se o e exes o a gi en Cse Γisdeno ed P oceedings o he 17 h Wo ld Cong ess The In e na ional Fede a ion o Au oma ic Con ol Seoul, Ko ea, July 6-11, 2008 978-3-902661-00-5/08/$20.00 © 2008 IFAC 15333 10.3182/20080706-5-KR-1001.3054 as e (Γ).The se BN⊂IRNdeno es he uni a yball BN={b∈IRN:kbk∞≤1}. 2.PROBLEMDESCRIPTION Conside he ollowing unce aindisc e e- imeLTIsys em: x+=Ax+Bu+w,y=Cx+Du(1) subjec o ollowing cons ain s: x∈X⊂IRnu∈U⊂IRm(2) and dis u bances se : w∈W⊂IRn(3) whe e: xis he cu en s a e, uis he cu en con ol ac ion, x+is he successo s a e, wisanunknowns a e dis u bance, y∈IRpis he cu en measu ed ou pu ,and (A,B,C,D)∈IRn×n×IRn×m×IRp×n×IRp×m.W, isa compac ,con ex no emp yse and X,Ua epolyhed al and poly opicse s espec i ely. Le φ(i;x,u,w)deno e he solu iono (1)a imeii he ini ials a eisxand he con oland dis u bance sequences a e, espec i ely,uand w. The o e all objec i eis os abilize he cons ained sys em and s ee he s a e o a neighbo hoodo he se poin ul illing he cons ain s o anypossibledis u bance. The ollowing s anding assump ionismade: Assump ion1.The couple(A,B)iscon ollable. 3.ROBUST MPCFOR TRACKING Fo he sakeo aclea and sel -con ained exposi iono he con ibu ions o he pape , he obus MPC o acking is succinc lyp esen ed in his sec ion. Amo ede ailed explana iono he con olle can be ound in(Al a ado e al. [2007], Al a ado. [2007]) 3.1Tube o ajec o ies The p oposed con olle isbased on he esponseo he nominalsys em,ob ained om(1)byneglec ing he dis u bances w.The nominalsys em isdesc ibed by: ¯x+=A¯x+B¯u,¯y=C¯x+D¯u(4) Choosing anini ials a e¯xand acon olsequence ¯ uyields as a esequence ¯ xob ained bysol ing (4) (¯xi=¯ φ(i;¯x,¯ u)). Tocoun e ac he dis u bances i isdesi able o o ce he ajec o y olieclose o he nominal ajec o y; hiscan bedone bychoosing he con olu osa is y: u=¯u+Ke e ∆ =(x−¯x) (5) whe eedeno es he e o be ween he s a eand he s a e o he nominalsys em.The e o edynamics isgi en by e+=AKe+w;AK=(A+BK) (6) I ma ixAKisHu wi z hen he eexis sa obus pos- i i elyin a ian se Z(Kolmano sky and Gilbe [1998], Rako ice al. [2005]) o he sys em (6) ha sa is ies AKZ⊕W⊆Z The in a iance p ope yallows os a e he no iono ube o ajec o ies gi en in he ollowing p oposi ion: P oposi ion1.(Mayne e al. [2005]).I he ini ial ealand nominalsys em s a es,sa is ye(0)=x(0)−¯x(0)∈Z, hen x(i)∈¯x(i)⊕Z∀i∈IN, o all dis u bance sequences w such ha w(i)∈W∀i=1,2,.... F om hisp oposi ioni isin e ed ha i he nominal con olac ions a ecalcula ed oensu e ha he nominal p edic ed s a es and inpu s sa is y he ollowing igh e cons ain s ¯ X=X⊖Z¯ U=U⊖KZ(7) hen he sys em con olled by(5)main ains all he possible ajec o ies admissible, ha is sa is ying he cons ain son s a es and inpu s o all possibleunce ain y(Mayne e al. [2005]). 3.2Se poin cha ac e iza ionandin a ian se o acking In his sec ion, asui ablepa ame iza iono he s eady s a es and inpu s o he nominalsys em isp esen ed. Conside a gi en ou pu a ge y , hen anys eady s a e o he nominalsys em zs=(xs,us)associa ed o his se - poin mus sa is y he ollowing equa ion A−InB0n,1 CD−Ip"xs us y #=0n,1 0p,1(8) Becauseo he pai (A,B)is s abilizable, he solu ion o hisp oblem can bepa ame e ized as zs=Mθθy =Nθθ(9) whe eθ∈IRnθisapa ame e ec o whichcha ac e izes anysolu ion, and Mθand Nθa esui ablema ices (Limon e al. [2008]). The exis ence o cons ain s(7)limi s he se o eachable s eady s a es and inpu s.The se o admissibles eady s a es isdeno ed asXsand i isapolyhed ongi en by Xs={xs∈¯ X:∃us∈¯ U|(A−In)xs+Bus=0n,1}. Now, he no iono in a ian se o acking isp esen ed De ini ion1.Le xebe he ex ended s a e(x,θ)∈ IRn+nθ, le KΩbeacon olgainsuch ha A+BKΩ isHu wi zand le Kθbegi en byKθ=[−KΩIm]Mθ. Then, ase Ωe ⊂IRn+nθisanin a ian se o acking admissible o (7), i o all (x,θ)∈Ωe , hen x∈¯ X, KΩx+Kθθ∈¯ Uand ((A+BKΩ)x+BKθθ,θ)∈Ωe . The oleo he p esen ed pa ame iza ionis osimpli y he p oposed p edic i econ olle aswellas he calcula iono in a ian se s o acking. 3.3P edic i e con olle The mainobjec i eo hiscon olle is o obus lys ee he ou pu sys em o a (neighbo hoodo )a a ge y . I isassumed ha his a ge can be cha ac e ized bya pa ame e ec o θbymeans o (9) (i.e. i isapossible s eady ou pu o he nominalsys em). Thiscon olle ,as he ube-based con olle (Mayne e al. [2005]),conside sasdecision a iables he ini ialnominal s a e¯xand he sequence o u u enominalcon olac ions ¯ u.Ino de odealwi h he se poin changep oblem an 17 h IFAC Wo ld Cong ess (IFAC'08) Seoul, Ko ea, July 6-11, 2008 15334 a i icials eady s a eand inpu (¯xs,¯us)=Mθ¯ θisconsi- de ed asdecision a iable. These a iables a ecalcula ed ominimize he ollowing nominalp edic ionbased cos unc ion: VN(x,θ;¯ u,¯x,¯ θ)= N−1 X i=0k¯x(i)−¯xsk2 Q+k¯u(i)−¯usk2 R +k¯x(N)−¯xsk2 P+k¯ θ−θk2 T(10) whe e¯x(i)=¯ φ(i;¯x,¯ u)and (¯xs,¯us)=Mθ¯ θ.The cu en s a exand he a ge ope a ion poin gi en byθa e pa ame e so he cos unc ion. Thiscos penalizes he de ia ion be ween he p edic ed ajec o yand he a i icials eady s a ealong he ho izon N.The de ia ion be ween he a i icials eady condi ions (gi en by¯ θ)and he a ge one (gi en byθ)ispenalized bymeans o he so-called o se cos k¯ θ−θk2 T. The cos unc ionmus beminimized conside ing he cons ain sde i ed oms abili yand admissibili ycondi- ions, leading o he ollowing quad a icp oblem PN(x,θ): min ¯x, ¯ u,θ VN(x,θ;¯ u,¯x,¯ θ) (11) s. .¯x∈x⊕(−Z) (12) ¯x(i)∈¯ X,i=0,···,N(13) ¯u(i)∈¯ Ui=0,···,N−1(14) (¯x(N),¯ θ)∈Ωe (15) whe e: ¯ X=X⊖Zand ¯ U=U⊖KZ. PN(x,θ)is sol ed online oyieldanop imal ini ials a e ¯x∗(x,θ),anop imalnominalcon olsequence ¯ u∗(x,θ)= {¯u∗(0,x,θ),¯u∗(1,x,θ),...,¯u∗(N−1,x,θ)}and he op imal a i icials eady condi ions pa ame e θ∗(x,θ).F om his, he con olapplied o he plan isgi en by: kN(x,θ)=¯u∗(0,x,θ)+K(x−¯x∗(x,θ)) (16) No ice ha he se o cons ain s ha de ines he easi- bili y egiono PN(x,θ)does no depend onθ( ha is, he a ge ope a ing poin ),bu onlyon he cu en s a e xand he decision a iables.The e o e, his easibili y egiononlydepends on he cu en s a ex.Thus,de ining ¯ XN⊂IRnas he se o he admissiblenominal ini ial s a es ¯xsuch ha (13),(14)and (15)hold, he se o easibles a es isgi en byXN=¯ XN⊕Z. 3.4S abilizing condi ionso he con olle Ino de oensu e obus s abili yo he closed loop sys em, he ollowing su icien condi ions on he de ining ing edien sa eassumed: Assump ion2.The ma ices Q,R,T,P,K,KΩ,and se s Ωe and Z ul il: •Q>0 and R>0 •The eexis sacons an σ>0such ha σT≥MT xMx, whe eMx=[In,0n]Mθ. •The gainma ixKis such ha A+BKisHu wi z. •The pai o ma ices KΩand Pa esuch ha A+BKΩ isHu wi zand P>0whe e P−(A+BKΩ)⊤P(A+BKΩ)=Q+K⊤ ΩRKΩ •Se Z⊂Xisanadmissible obus posi i elyin a ian se such ha (A+BK)Z⊕W⊆Zand KZ⊂U. •Ωe isanin a ian se o acking (asla geaspossible) o he nominalsys em (4)subjec o he cons ain s ¯ Xand ¯ Uand using ascon olgainma ixKΩ. Nowi ispossible oes ablish he ollowing esul : Theo em1.(Al a ado e al. [2007]).Conside sys em (1) subjec o he cons ain s(2)and such ha e i ies As- sump ions 1 and 2.Le kN(x,θ)be he con ol law esul - ing om he solu iono he op imalp oblem PN(x,θ).Le Y be he se o eachable a ge ou pu sgi en by Y ∆ ={y=Nθθ:Mθθ∈P ojx(Ωe )ׯ U} Then ∀x(0)∈XNand a a ge y ∈Y he closed loop sys em ul ils he cons ain s(2)along i se olu ionand con e ges asymp o ically oxs⊕Z. The p oo o his heo em can be ound in(Al a ado. [2007]) 3.5Cancella iono heou pu o se The p oposed con olle isables ee he sys em s a e o aneighbo hoodo he a ge s eady s a e, ha isxs⊕Z. I he dis u bances a edecaying, he sys em con e ges asymp o ically o he a ge ,bu i he dis u bance ends o a s eady alue, he he closed loopsys em p esen so se on he ou pu s, i.e. he e o y(k)−y ends o a cons an alue. Thiscan becancella ed by he ollowing me hods.One possible echnique isaugmen ing he model o he sys em wi hin eg a ing dis u bances o emo eo se (Pannocchia [2004]);alinea con ol lawensu es obus o se - ee con- ola expenseo enla ging h ee imes he o de o he plan whichmakes he con olle designand calcula ion mo edi icul . Asecond me hodp oposed by he au ho sin(Al a ado e al. [2007]),achie es he o se ee wi hou augmen ing he plan ,bu adding anou e loopconsis ing o adis u - bance es ima o and ase poin co ec iongi en by ˆy (k)=y −(C+DK)(In+(A+BK))−1ˆw(k) whe eˆw(k)is he es ima ed dis u bance. The es ima o dynamics should beadjus ed acco ding he dynamics o he dis u bance signaland he closed loopsys em. 4.SYNTHESISOF THE PROPOSEDCONTROLLER The p esen ed p edic i econ olle isdesigned bypicking he de ining pa ame e s(ho izon, ma ices and se s) ul ill- ing Assump ion2.Since hisdoes no ix he pa ame e s, hesecan be chosen acco ding ocon olobjec i es,such asclosed loop pe o mance, dis u bance ejec ion, domain o a ac ion, e c. Ma ices Qand Rde ine he s agecos whichisclosed loop pe o mance index.Con olho izonNis ypicallychosen asla geaspossible, since ala geNp o ides ala ge domaino a ac ionand anenhancemen o he closed loop pe o mance. Howe e , he dimensiono he p oblem g owswi hN,and hence equi es ala ge compu a ional ime. The e o e, a adeo mus beachie ed. 17 h IFAC Wo ld Cong ess (IFAC'08) Seoul, Ko ea, July 6-11, 2008 15335 In his sec ion, he chosen o he ex apa ame e s speci ic o he p oposed p edic i econ olle : he ma ices T,K, KΩ,Pand he se sZand Ωe . 4.1Thepa ame e T Thisma ixde ines he weigh o he acking e o cos k¯ θ−θk2 Tin he cos unc ion. The e ec so hispa ame e on he closed loopsys em a e he ollowing : •T ansien dynamicsandse poin il e ing: i ma ix Tischosen openalize mo ehea ily he acking e o cos , hen he con e gence o ¯ θ oθismade as e , and hence he ansien o he closed loopsys em o he a ge . •Localop imali y: i iswellknown ha MPCcan be designed obelocallyin ini eho izonop imal( o he nominalsys em),howe e he addi iono he a i icial s eady s a eand inpu makes ha hisp ope ydoes no holdin he p esen ed con olle .The op imali y loss can bea bi a ily educe bypicking ala ge enoughma ixT. •O se minimiza ion: hisisa ema kablep ope y o he con olle (Al a ado. [2007]).Conside ha he a ge y isno eachable ha is, he desi ed ou pu y does no ul il anyo he cons ain s, i.e. Mθθ6∈ Ω ׯ U,∀θ:y =Nθ.Unde hesecondi ions, he a ge canno be eached and hen, he ou pu will p esen o se . In hiscase he p oposed con olle d i es he sys em o he neighbo hood˜xs⊕Z,whe e he s eady s a e˜xs (gi en by he pa ame e ˜ θ)is he one whichminimizes he o se cos .Tha is ˜ θ=a gmin ¯ θk¯ θ−θk2 T Mθ¯ θ∈Ω ׯ U The e o e, ma ixTallowsus op io i ize some ou pu s(byweigh ing mo ehea ilyi sco esponding e minma ixT) o achie eaminimum o se on hisou pu s.See ha hisp io i iza iondoes no a ywi h he scaling o ma ixT,and hence he e ec sp e iouslyp esen ed can besimul aneously conside ed in he designo T. Then he sensibleway odesignTis i s lypick he s uc u eacco ding o he o se minimiza ion, and hen, scale he ma ix o achie eaquick ansien wi hasmall enoughop imali yloss.Finallyno ice ha since he alue o Tisindependen o he es o pa ame e s, hiscan be uned online main aining he s abilizing p ope ies. 4.2Thepa ame e sKΩ,PandΩe The e minalcos and e minalse s onglydepend on he con olgainKΩ.Thiscan bedesigned acco ding he ollowing aspec s: omape o mance poin o iew, i is desi able ha he e minalcos is aken as he cos - o-go, whichcan beensu ed i KΩischosen as he LQ egula o gain. F omadomaino a ac ion iewpoin , i isdesi able o akeKΩ o achie eala ge e minalse Ωe . In hiscase, he la gesize o he in a ian se o acking p o ides ala gedomaino a ac ionXN,e en o small alues o he con olho izon. Then, he mos sensibleway opickKΩis he Linea Quad a icRegula o and he ma ixPas he solu iono he Ricca iequa ion. The in a ian se o acking Ωe can be aken asapoly- hed alse whichapp oxima es a bi a ily o he maximal in a ian se o acking hanks o a simplep ocedu e (Limone al. [2008]. Thisp o ides ase o eachable a ge sY p ac icallyequal o he maximalone (i.e. he di - e ence be ween bo hse scan bemade a bi a ilysmall). 4.3Thepa ame e K Thispa ame e hasanimpo an olein he p oposed con olle since hiscon olgainisused ocompensa e he de ia ion om he nominalp edic ions in he con olle (16),and he e o e, i cha ac e izes he dynamics o he closed loopsys em in he p esence o dis u bances.This makes ha Kischosen acco ding o a obus nesso dis u bance ejec ionc i e ium. In hispape ,weconside as obus nessc i e ium he minimiza iono he minimum admissible obus posi i ely in a ian se .Thus,Kischosen o:(i)ensu e he exis ence o anadmissible obus posi i elyin a ian se Zsuch ha he se sX⊖Zand U⊖KZa eno emp yse s;(ii) minimize he size o Z.Thisc i e ium p o ides ala ge domaino a ac ionand ala ge se o eachable a ge s aswellasminimizes he e ec o he dis u bance on he ajec o yo he sys em. Conside w.l.o.g. ha se X={x:|h⊤ ix|≤1,i= 1,···,n x}and se U={u:|ℓ⊤ ju|≤1,j=1,···,n u}. Then, he syn hesisp oblem osol eis ocalcula e he con ol lawu=Kxsuch ha he size o he ellipsoid E(P,1)={x∈IRn:x⊤Px≤1}isminimized ul illing ha : (i)E(P,1)isa obus in a ian se o sys em (6).This condi ioncan be e o mula ed as ollows: (x+)⊤Px+≤1,∀x∈E(P,1),∀w∈W(17) Applying he S-p ocedu e, and conside ing he con- exi ywi h ela ion ow,equa ion(17)is sa is ied i exis sλ≥0such ha : ((AK)x+w)⊤P((AK)x+w)+λ(1−x⊤Px)<1 AK=A+BK∀x∈IRn,∀w∈ e (W) (18) whe e e (W)deno es he se o e exes o W.This can be ew i en as: λP−(A+BK)⊤P(A+BK)−(A+BK)⊤Pw −w⊤P(A+BK)1−λ−w⊤Pw>0 ∀w∈ e (W) (ii)Fo all x∈E(P,1), he con ol law|ℓ⊤ jKx|≤ρj o all j=1,···,n uand ρj∈(0,1]. The oleo he pa ame e ρjis o es ic he se o admissiblecon ol inpu s o gua an ee a gi en con ol angeo he he MPCcon olle i.e. he se ¯ U=U⊖KZhasno emp yin e io . Thiscondi ioncan beposed as: ℓ⊤ jKP−1K⊤ℓj≤ρ2 j,j=1, . . . , n u Applying he Schu complemen , hisyields o ρ2 jℓ⊤ jK K⊤ℓjP>0,j=1, . . . , n u 17 h IFAC Wo ld Cong ess (IFAC'08) Seoul, Ko ea, July 6-11, 2008 15336 (iii)Fo all x∈E(P,1),|h⊤ ix|≤1.Conside ing simila a gumen s o he p e ious ac , hiscondi ionis equi alen o 1h⊤ i hiP>0,i=1,...,n x Ino de ominimize he size o he ellipsoidE(P,1), asui ablemeasu eo his se mus be chosen. In his pape ,wep oposeasmeasu eapa ame e γ>0such ha E(P,1)⊆√γX.The e o e, minimize he size o E(P,1)isposed asminimizing he pa ame e γ.Ob iously, admissibili yo he solu ion equi es ha γ≤1. Applying s anda dope a ions o LMIs(Boyde al. [1994]), he p oposed syn hesisp ocedu ecan be o mula ed as he solu iono he ollowing con ex op imiza ionp oblem: min Y,W,γγ s.a. "λW∗ ∗ 0 1 −λ∗ AW+BYwW#>0,∀w∈ e (W) ρ2 i∗ Y⊤ℓiW>0,i=1,...,n u γ∗ WhiW>0,i=1,...,n x o a gi en λ≥0.I easible, he ellipsoidisgi en by P=W−1and he con olgainisK=YW−1.I iswo h ema king ha any obus c i e ium ha can beposed as LMIscan beadded o his syn hesisp oblem. 4.4Calcula iono he obus in a ian se Z Once he con olgainKisdesigned, anadmissible obus posi i elyin a ian (RPI)se (as small aspossible) mus becalcula ed. See ha he p oposed syn hesiso Kensu es he exis ence o his se .I would bedesi able ocompu e he minimum obus posi i elyin a ian se (mPRI)F∞ (Kolmano sky and Gilbe [1998], gi en by Fs= s M k=0 (A+BK)kW when s ends oin ini y.Un o una ely,F∞canonlybe calcula ed o somespecialcases,suchas,when adead- bea con ol lawisused. In he ecen pape (Rako ice al. [2005], ap ocedu e o he de e mina iono anin a ian app oacho F∞is p esen ed. Thisallowsone ocompu eaRPIZsuch ha F∞⊆Z⊆F∞⊕ǫBn o a gi en bound o he absolu e e o ǫ.To hisaim, he ollowing unc ions a ecalcula ed α(s)=minα:(A+BK)sW⊆αW β(s)=minβ:Fs⊆βBn These alues a ecalcula ed bysol ing anumbe o Linea P og amming p oblems.I ala geenough alue o sis aken such ha (1−α(s))−1α(s)β(s)≤ǫ hen he se Z=(1−α(s))−1Fsisanapp oxima iono F∞wi han e o bound less hanǫ. F omap ac icalpoin o iew, i migh bedesi able o calcula e he app oxima ion o a ela i ee o bound, whichdoes no equi eanap io ies ima iono he size o he in a ian se ochoose he e o bound. Thisis s a ed in he ollowing lemma. Lemma 1.Le sbeaposi i e ealnumbe such ha α(s)≤λ 1+λ o a gi en ela i ee o bound λ∈(0,1).Then he se Z=(1−α(s))−1FsisaRPIsuch ha F∞⊆Z⊆(1+λ)F∞ On he o he hand, wespecialize his esul o he case ha he unce ain yse isazono opeo he o m W=HBn⊕w0 whe eH∈IRn×nisnonsingula .Thisismo i a ed be- cause hisclass o se is equen lyused obound addi i e unce ain ies inp ac ice. Fo hisclass o unce ain ies we cans a e he ollowing lemma. Lemma 2.Conside ase W=HBn⊕w0whe eHisa nonsingula ma ix.Deno e(A+BK)asAKand de ine he ma ixHz(s)=[As−1 KH,As−2 KH,···,H]. Then kH−1As KHk∞=minα:As KHBn⊆αHBn kHz(s)k∞=minβ: s−1 M k=0 Ak KHBn⊆βBn Based on hislemma,anapp oxima ion o he mRPI wi ha ela i ee o boud λ o he caseo zono opic unce ain ies isp esen ed in he ollowing lemma. Lemma 3.Le sbesuch ha kH−1(A+BK)sHk∞≤λ 1+λ o a gi en ela i ee o bound λ∈(0,1)and deno e ˆα(s)=kH−1(A+BK)sHk∞.Then he zono opeZgi en by Z=(1−ˆα(s))−1Hz(s)Bsn⊕(In−(A+BK))−1w0 isaRPIsuch ha F∞⊆φK⊆(1+λ)F∞. The p oo so heselemmascan be ound in([Al a ado., 2007,Chap e 6]. The zono opeexp essiono he in a ian se makes easie he calcula iono he linea mapping and Pon yagindi - e ence, and consequen ly he calcula iono he poly opes X⊖Zand U⊖KZ.Howe e ,cons ain (12) equi es he calcula iono he se o inequali ies (hype planes) ha de ine Z.Azono opeisacompac exp essiono apoly ope wi hanumbe o ace s ha g owsexponen iallywi h he dimensiono he uni a ybox(sn).Al hough he e exis s specialized algo i hms o ob ain he ace s, hesea e only ac able o a educed dimensiono he zono ope. In([Al a ado., 2007,Chap e 6])can be ound se e al p ac icalp ocedu es o hiscalcula ion. 5.ILLUSTRATIVE EXAMPLE Conside acons ained sampled doublein eg a o 17 h IFAC Wo ld Cong ess (IFAC'08) Seoul, Ko ea, July 6-11, 2008 15337 x+=1 1 0 1 x+0 0.5 1 0.5u+w y=[0 1 ]x. whe e he dis u bances a ebounded inw∈W=0.1B2. The sys em mus ul il he ollowing cons ain s:|x1|≤5, |x2|≤5,|u1|≤0.3,|u2|≤0.3. The objec i eis oshow he p oposed p ocedu es o ind asui able obus con olle gainK oensu e he admissibili yo he ube-based p edic i econ olle , i.e. ensu ing ha X⊖Zand U⊖KZa eno emp yse s. To es he p oposed syn hesisme hod, wedesign wo ma ices,KLQRand Kop.The i s ma ixisob ained as he LQR gain o Q=I2and R=10I2.The second ma ix isdesigned using he p oposed me hodwi ha alue o ρ=0.48 ino de o ge he samese KZasin he p e ious case. Figu e1shows he app oxima ed RPI o bo hgains. I can beseen ha he p oposed me hodp o ides asmalle se Z. −0.8 −0.6 −0.4 −0.2 00.2 0.4 0.6 0.8 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 x1 x2 Compa a ion o he minimal obus in a ian se s −0.3 −0.2 −0.1 00.1 0.2 0.3 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 u1 u2 Con ol cons ain s KZLQR KZop U _ _ ULQR Uop Zop ZLQR Fig.1.Compa a i eo he admissibleminimal obus in a ian se s Ino de odemons a e he oleo he pa ame e ρ, le us y ocalcula e he minimum obus in a ian se wi h he minimum size, o ha ,pickaρ=1 and conside a LQR gainwi hQ=1000 ∗I2and R=1.Figu e2shows he RPIde i ed omLQR and he p oposed me hod. I can beseen how he se KZLQRisno easibledue o iola es he con olcons ain s(¯ ULQR=U⊖KZLQRis anemp yse ).The p oposed me hodp o ides a easible RPIbu ,since ρ=1, he se ¯ Uop=U⊖KZopisqui e small, sosmalle is he pa ame e ρbigge is he obus minimal in a ian se (wo s dis u bance ejec ion) bu bigge is¯ Uop(The nominalsys em hasaless es ic i e con olcons ain s,so he e olu iono he nominalsys em is as e ).Thus,wi h he alue o ρi ispossible o choosebe ween dis u bance ejec ionand pe o mance, as bigge isρbe e dis u bance ejec ionbu wo seis he pe o mance. 6.CONCLUSIONS Thispape dealswi h he designs ep o he obus p edic i econ olle o acking (Al a ado e al. [2007]. Thiscon olle can be hough asana u alex ensiono p edic i econ olle s o achie ese poin acking and has −0.2 −0.15 −0.1 −0.05 00.05 0.1 0.15 0.2 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 x1 x2 Compa a ion o he minimal obus in a ian se s −0.3 −0.2 −0.1 00.1 0.2 0.3 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 u1 u2 Con ol cons ain s ZLQR KZop Zop U Uop _ KZLQR Fig.2.Compa a i eo he minimal obus in a ian se s someex apa ame e s ha can be uned o achie e he ollowing objec i es: •O se minimiza ion •Dis u bance ejec ion byminimizing he RPI •Gua an ee o easibili yo he ube-based MPCand enla gemen o he domaino a ac ion. The i s one depends on he acking e o cos weigh ing ma ixT,whichcan be eelychosen (unde amild assump ion) openalize mo ehea ilysomeou pu sin o de ominimize hei o se .The las wo objec i es a e achie ed bymeans aLMIwhichcan bee icien lysol ed. Finally,wep esen ame hod oes ima e he mRPIin he case ha ela i ee o bound isused, and he unce ain y isposed asaclass o zono opes.Wealsop o ide p ac ical me hods ode i e eliableapp oxima ions incaseo exac calcula ionisno a o dable. REFERENCES I.Al a ado.MPC o T ackingo Cons ainedLinea Sys ems.PhD hesis,Uni o Se ille.,2007. I.Al a ado,D.Limon, T.Alamo,M.Fiacchini,and E.F. Camacho.Robus ubebased MPC o acking o piece- wisecons an e e ences.InP oceedingso heCDC, 2007. S. Boyd, L.E.Ghaoui,E.Fe on, and V.Balak ishnan. Linea Ma ixInequali iesinsys emsandcon ol heo y. SIAM,1994. I.Kolmano sky and E.G.Gilbe .Theo yand conm- pu a iono dis u bance in a ian se s o disc e e- ime linea sys ems.Ma hema icalP oblemsinEnginee ing: Theo y,Me hodsandApplica ions,4:317–367,1998. D.Limon, I.Al a ado,T.Alamo,and E.F.Camacho. MPC o acking o piece-wisecons an e e ences o cons ained linea sys ems.Au oma ica,2008.Accep ed o publica ion. D.Q. Mayne, M.M.Se on, and S.V.Rako ic. 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