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)
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(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
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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.
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17 h IFAC Wo ld Cong ess (IFAC'08)
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15338