Escuela T´ecnica Supe io de Ingenie ´ıa
Depa amen o de Ingenie ´ıa de Sis emas y Au om´a ica
Doc o al Thesis
P obabilis ic da a-d i en
me hods o o ecas ing,
iden i ica ion and con ol
Al onso Daniel Ca ne e o Pandu o
Supe ised by:
Daniel Rod ´ıguez Ram´ı ez
Teodo o ´
Alamo Can a e o
Se ille, Sep embe 2022
Con en s
Acknowledgemen s i
Abs ac iii
No a ion, con en ions and de ini ions
1 In oduc ion 1
1.1 Mo i a ion and objec i es . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 The p oblem o sys em iden i ica ion . . . . . . . . . . . . . 1
1.1.2 Quan i ying he unce ain y . . . . . . . . . . . . . . . . . . 4
1.1.3 Model p edic i e con ol . . . . . . . . . . . . . . . . . . . . 7
1.1.4 Objec i es o his disse a ion . . . . . . . . . . . . . . . . . 11
1.2 Ou line ................................. 11
1.3 Publica ions............................... 13
I P obabilis ic o ecas ing 15
2 Fo ecas ing using dissimila i y unc ions 17
2.1 P oposed dissimila i y unc ion . . . . . . . . . . . . . . . . . . . . 18
2.1.1 Cla i ying example . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Dissimila i y unc ions and eg ession . . . . . . . . . . . . . . . . 22
2.3 Applica ion: o ecas ing s ock p ices using dissimila i y unc ions . 26
2.3.1 Resul s ............................. 27
2.4 Conclusions............................... 31
3 P obabilis ic p edic ion egions 33
3.1 Uni a ia ecase ............................. 33
3.1.1 Empi ical p obabili y densi y unc ion . . . . . . . . . . . . 34
3.1.2 Cla i ying example: uni o m dis ibu ion . . . . . . . . . . . 38
3.1.3 Nume ical example: Lo enz a ac o . . . . . . . . . . . . . 38
3.1.4 Nume ical example: Dow Jones indus ial a e age index . . 40
3.2 Mul i a ia ecase............................ 44
3.2.1 Implici egions . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.2 Cla i ying example: mul i a ia e uni o m dis ibu ion . . . 46
3.2.3 Ellipsoidal p edic ion egions . . . . . . . . . . . . . . . . . 46
3.2.4 Nume ical esul s . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Conclusions............................... 52
II K iging-based iden i ica ion 53
4 S a e-space k iging o au onomous sys ems 55
4.1 Dynamick iging ............................ 55
4.2 Linea s a e-space k iging . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.1 Ini ial condi ion . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.2 Local-da a app oach . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Ke nel-based s a e-space k iging . . . . . . . . . . . . . . . . . . . 60
4.3.1 Ini ial condi ion . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Kalman il e o SSK . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5 Nume ical examples . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.5.1 Sunspo numbe . . . . . . . . . . . . . . . . . . . . . . . . 65
4.5.2 R¨ossle a ac o . . . . . . . . . . . . . . . . . . . . . . . . 66
4.6 Conclusions............................... 68
5 S a e-space k iging o non-au onomous sys ems 69
5.1 Non-au onomous linea SSK . . . . . . . . . . . . . . . . . . . . . . 69
5.1.1 Ini ial condi ion . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Non-au onomous ke nel-based SSK . . . . . . . . . . . . . . . . . . 71
5.2.1 Ini ial condi ion . . . . . . . . . . . . . . . . . . . . . . . . 72
5.3 Applica ion oMPC .......................... 73
5.3.1 Nominal s abili y analysis . . . . . . . . . . . . . . . . . . . 75
5.3.2 Robus s abili y analysis . . . . . . . . . . . . . . . . . . . . 76
5.4 Examples ................................ 77
5.4.1 Con inuously-s i ed ank eac o . . . . . . . . . . . . . . . 78
5.4.2 Tempe a u e con ol lab . . . . . . . . . . . . . . . . . . . . 79
5.5 Conclusions............................... 81
III P obabilis ically-ce i ied da a cen e managemen 87
6 Bounds on he cons ain iola ion le el 89
6.1 In oduc ion............................... 89
6.2 Main esul s............................... 90
6.2.1 A i s bound on cons ain iola ion a e . . . . . . . . . . 91
6.2.2 A di e en bound . . . . . . . . . . . . . . . . . . . . . . . 92
6.3 Cla i yingExample........................... 94
6.4 Conclusions............................... 97
7 Ene gy-e icien managemen o da a cen e s 99
7.1 In oduc ion............................... 99
7.2 Da a cen e desc ip ion . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2.1 Tasksmodel...........................102
7.2.2 Se e model ..........................102
7.2.3 The malmodel.........................104
7.2.4 Quali y o se ice . . . . . . . . . . . . . . . . . . . . . . . . 106
7.3 Managemen app oach . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.4 Pa icle based sol e s o complex op imiza ion p oblems . . . . . . 108
7.4.1 Scena io-based app oach . . . . . . . . . . . . . . . . . . . . 111
7.4.2 Pa allel implemen a ion . . . . . . . . . . . . . . . . . . . . 113
7.5 Bounds on he cons ain iola ion a e . . . . . . . . . . . . . . . 114
7.6 Nume ical esul s............................ 114
7.6.1 QoS iola ion a e . . . . . . . . . . . . . . . . . . . . . . . 116
7.6.2 The mal cons ain iola ion a e . . . . . . . . . . . . . . . 117
7.6.3 Pa allel compu a ion imp o emen . . . . . . . . . . . . . . 118
7.6.4 Compu a ion ime analysis . . . . . . . . . . . . . . . . . . 119
7.7 Conclusions...............................120
8 Conclusions and u u e wo k 121
8.1 Con ibu ions..............................121
8.2 Fu u ewo k...............................122
Bibliog aphy 125
Acknowledgemen s
Me oy a oma la libe ad de esc ibi es a secci´on en cas ellano debido a que
algunas de las pe sonas a las que a di igida no ienen un conocimien o luido de
la lengua de Shakespea e y, po an o, odo es e es ue zo pod ´ıa esul a en ano.
Adem´as que, despues de esc ibi nume osas p´aginas en igu oso ingl´es, no es ´a de
m´as ol e a las p opias a´ıces po un iempo.
Han pasado ap oximadamen e 9 a˜nos desde que en ´e a es udia en la escuela de
ingenie os y, has a cie o pun o, pa ece como si ue a aye (lo cual es p obable-
men e el ´opico m´as usado de la his o ia). El c´omo llegu´e aqu´ı me pa ece, en cie o
sen ido, pu a casualidad, pues du an e mi m´as ie na in ancia y adolescencia, mis
in e e es labo ales y p o esionales a ia on con ecuencia. En cualquie caso, c eo
que debo g an pa e de mi in e ´es a la ingenie ´ıa y al con ol a An onio Nue o y
a Juanma Esca˜no. Si no hubie a ecuen ado aquel alle de au oma izaci´on en el
Colegio Al ai cuando me encon aba en P ima ia, p obablemen e mi ayec o ia
hubie a sido dis in a.
Po o o lado, debo a mi he mano Jos´e Ma ´ıa mi in e ´es po la ca e a in es i-
gado a, la cual ´el comenz´o mucho an es que yo en o a acul ad de es a misma
uni e sidad. Po alg´un mo i o, pens´e que se a aba de un abajo ascinan e con
una calidad de ida bas an e buena. En o o o den de cosas, mi pad e esul ´o se
un g an apoyo du an e mis p ime os pasos en la escuela, debido a las expe ien-
cias simila es que hab´ıa su ido du an e su ju en ud. En cuan o a mis he manas,
Ana y Guadalupe, les debo el no es i como un po diose o (lo cual no es poco).
Bueno, algo m´as hab ´a, pe o eso es lo p ime o que se me ha enido a la men e.
Pa a mi mad e, s´olo se me ocu e deci que es ´unica en odos los sen idos.
Tambi´en me gus a ´ıa ag adece especialmen e a Se gio Luc´ıa po su acogida en
la Uni e sidad T´ecnica de Do mund du an e mi es ancia de doc o ado. Fue una
de esas expe iencias posi i as en las que uno descub e que el es o del mundo no
es como Se illa (quien enga o´ıdos que oiga).
Po supues o, y como no pod´ıa se de o a mane a, ambi´en ag adezco la labo
que han enido mis di ec o es Dani y Teo en es e abajo y po la o maci´on
ecibida a lo la go de odo es e iempo. Aunque nues os caminos se sepa en
empo almen e, espe o que uel an a jun a se en un u u o ce cano. Tambi´en,
me gus a ´ıa ag adece especialmen e a Dani Lim´on po la opo unidad que me
b ind´o en su momen o. G acias a eso, pude empeza a abaja en es e g upo de
in es igaci´on. A su ez, me gus a ´ıa ag adece a los compa˜ne os de doc o ado que
me ayuda on en mis comienzos, especialmen e a Pablo y Pepe. De mane a simila ,
engo ambi´en un hueco pa icula pa a mis compa˜ne os de ca ´es y/o desayunos
Joaqu´ın y Jos´e An onio.
Po o a pa e, exis en a su ez g an can idad de pe sonas que han con ibuido
i
in ini esimalmen e a con e i me en lo que soy a d´ıa de hoy y a las que, po an o,
es oy muy ag adecido. No quie o ealiza una enume aci´on exhaus i a po que eso
esul a sumamen e pelig oso (la p obabilidad de ol ida a alguien es bas an e al a
y no es di ec amen e p opo cional a la impo ancia de la pe sona en cues i´on).
Po an o, si es ´as leyendo es a abu ida secci´on, puedes conclui sin emo a
equi oca e que e es una de esas pe sonas.
Po ´ul imo, y no po ello menos impo an e, me gus a ´ıa dedica es a esis a mis
o os dos he manos, Juanma y Albe o, los cuales no c eo que engan un in e ´es
especial po es a dedica o ia. En cualquie caso, ah´ı queda dicho.
Al onso Daniel Ca ne e o Pandu o,
Se illa, sep iemb e de 2022.
ii
Abs ac
This disse a ion p esen s con ibu ions mainly in h ee di e en ields: sys em
iden i ica ion, p obabilis ic o ecas ing and s ochas ic con ol.
Thanks o he concep o dissimila i y and by de ining an app op ia e dissimi-
la i y unc ion, i is shown ha a amily o p edic o s can be ob ained. Fi s , a
p edic o o compu e nominal o ecas ings o a ime-se ies o a dynamical sys em
is p esen ed. The e ec i eness o he p edic o is shown by means o a nume ical
example, whe e daily p edic ions o a s ock index a e compu ed. The ob ained
esul s u n ou o be be e han hose ob ained wi h popula machine lea ning
echniques like Neu al Ne wo ks.
Simila ly, he a o emen ioned dissimila i y unc ion can be used o compu e condi-
ioned p obabili y dis ibu ions. By means o he ob ained dis ibu ions, in e al
p edic ions can be made by using he concep o quan iles. Howe e , in o de o
do ha , i is necessa y o in eg a e he dis ibu ion o all he possible alues o
he ou pu . As his nume ical in eg a ion p ocess is compu a ionally expensi e,
an al e na e me hod bypassing he compu a ion o he p obabili y dis ibu ion is
also p oposed. No only is compu a ionally cheape bu i also allows o compu e
p edic ion egions, which a e he mul i a ia e e sion o he in e al p edic ions.
Bo h me hods p esen be e esul s han o he baseline app oaches in a se o
examples, including a s ock o ecas ing example and he p edic ion o he Lo enz
a ac o .
Fu he mo e, new me hods o ob ain models o nonlinea sys ems by means o
inpu -ou pu da a a e p oposed. Two di e en model app oaches a e p esen ed:
a local da a app oach and a ke nel-based app oach. A kalman il e can be added
o imp o e he quali y o he p edic ions. I is shown ha he o ecas ing pe o -
mance o he p oposed models is be e han o he machine lea ning me hods in
se e al examples, such as he o ecas ing o he sunspo numbe and he R¨ossle
a ac o . Also, as hese models a e sui able o Model P edic i e Con ol (MPC),
new MPC o mula ions a e p oposed. Thanks o he dis inc i e ea u es o he
p oposed models, he nonlinea MPC p oblem can be posed as a simple quad a ic
p og amming p oblem. Finally, by means o a simula ion example and a eal
expe imen , i is shown ha he con olle pe o ms adequa ely.
On he o he hand, in he ield o s ochas ic con ol, se e al me hods o bound
he cons ain iola ion a e o any con olle unde he p esence o bounded o
unbounded dis u bances a e p esen ed. These can be used, o example, o une
some hype pa ame e s o he con olle . Some simula ion examples a e p oposed
in o de o show he unc ioning o he algo i hms. One o hese examples con-
side s he managemen o a da a cen e . He e, an ene gy-e icien MPC-inspi ed
iii
4Chap e 1. In oduc ion
Combina ions o some o he a o emen ioned me hods ha e been explo ed as well,
i.e. neu o- uzzy me hods [37, 38] combining Takagi-Sugeno models and Neu al
Ne wo ks. Recen ly, ano he echnique ha is ge ing inc easing a en ion is he
Koopman ope a o [39, 40], whe e he nonlinea dynamics o he sys em a e con-
e ed in o linea dynamics in a in ini e-dimensional s a e ec o . Al hough he
sys em becomes in ini e-dimensional, i is possible o app oxima e hem by a su -
icien ly la ge s a e- ec o and apply me hods om linea con ol heo y.
All o hese me hods assume ha he dynamics o he sys em a e comple ely
unknown. Howe e , i may exis si ua ions whe e some pa ial knowledge o
he sys em can be used alongside some black-box model. This leads us o he
hyb id modeling amewo k [41, 42]. He e, he ou pu s a e compu ed as he
sum o wo e ms, one co esponding o he known dynamics and ano he one
co esponding o he unknown dynamics. No e ha his is e y di e en o he
g ey-box modeling app oach men ioned be o e.
Figu e 1.2: F om i s -p inciples models o black-box models.
1.1.2 Quan i ying he unce ain y
Howe e , some imes we a e in e es ed no only in ob aining nominal p edic ions
o a ce ain sys em, bu also a quan i ica ion o hei unce ain y. Tha is, when
he conside ed sys em p esen s noisy measu emen s and/o addi i e dis u bances,
i would be use ul o cha ac e ize a egion whe e he eal ou pu o he sys em
may be wi h a speci ied p obabili y ins ead o only compu ing he expec ed alue.
Fi s , we ocus on he uni a ia e case, i.e. in e al p edic ions. Gi en he eg esso
k, he objec i e is o compu e an in e al I( k)=[y−
k, y+
k] such ha we maximize
he p obabili y ha ykbelongs o I( k) while minimizing he in e al wid h (y+
k−
y−
k). These wo con lic ing objec i es can be econciled i one minimizes he
in e al wid h subjec o he cons ain ha I( k) con ains ykwi h a p e-speci ied
p obabili y.
In e al p edic ions play a ele an ole in he con ol o unce ain sys ems. Zono-
opes and DC P og amming a e used o ob ain in e al s a e es ima o s in [43]
and [44] espec i ely. In e al obse e s o linea ime- a ying sys ems ha e been
p oposed in [45] and [46]. Faul de ec ion me hods based on zono opic bounds
1.1. Mo i a ion and objec i es 5
can be ound in [47]. In [48], se heo e ic app oaches a e also used in he con ex
o aul de ec ion. Se membe ship me hods [49, 50] can also be used o ob ain
in e al p edic ions. A mixed Bayesian/se -membe ship app oach is p oposed in
[51].
The e exis s di e en me hods in he li e a u e ha add ess he p oblem o ob-
aining in e al p edic ions. Fo example, i he ec o o unce ain y is bounded
and he conside ed sys em sa is ies some Lipschi z assump ions, one can eso o
bounded e o me hods [52] ha gua an ee ha ykis always con ained in I( k).
See, o example, [53] and [27]. O he bounded e o s a egies ha e been p o-
posed in [54, 55, 56]. The s a is ical cha ac e iza ion o noise and dis u bances
can be used o enhance he pe o mance o in e al es ima ion me hods. See
[57, 58, 59] and e e ences he ein. Also, p obabilis ic alida ion me hods can be
used o assess he pe o mance o he in e al p edic o s [60, 61, 62, 63].
An impo an concep is ha o quan iles [64, 65]. Deno e Fyk(a| k) as he cumu-
la i e dis ibu ion unc ion o he associa ed ou pu ykcondi ioned o he eg esso
k, ha is,
Fyk(a| k) = P ob{yk≤a| k},
whe e ais a scala . Gi en k, we say ha aτis he condi ioned τ-quan ile i
Fyk(aτ| k) = P ob{yk≤aτ| k}=τ.
The es ima ion o he condi ioned quan iles is ele an in mul iple applica ions
(see [66] and [67]) and can be add essed using di e en me hodologies. The mos
classical app oach elies on he assump ion ha ykand ka e join ly no mal. Tha
is, he assump ion ha he (join ) p obabili y densi y unc ion o he ( andom)
a iables yand is a mul i a iable no mal p obabili y densi y unc ion. Unde
his assump ion, he condi ioned p.d. . is a mono a iable no mal p.d. . and
he quan iles can be ob ained in a simple and di ec way [68]. Un o una ely,
he me hods based on no mal dis ibu ions a e e y sensi i e o he p esence o
ou lie con amina ion. Mo eo e , in many long- ailed dis ibu ions, he no mal
assump ion is no well sui ed o cha ac e ize con idence in e als and one has
o eso o non-Gaussian dis ibu ions. In hese cases, gene aliza ions o he
Chebyshe inequali y can be used o ob ain p obabilis ic bounds [69, 70].
The compu a ion o he condi ioned quan iles can be also add essed by means o
pa ame ic eg ession echniques [65], [66]. Assuming ha he e exis s θ o which
yk≈θ⊤ k, hen pa ame e ec o θcan be chosen as he one ha minimizes a cos
unc ion o he e o θ⊤ k−yk. I one chooses a cos unc ion ha penalizes in an
asymme ic way posi i e and nega i e e o s hen a quan ile eg esso is ob ained.
Gi en he aining pai s (¯yi,¯ i), i= 1, . . . , N and τ∈(0,1), he quan ile eg esso
is de ined in e ms o he ollowing op imiza ion p oblem
min
θ
N
X
i=1
(1 −τ) max{0, θ⊤¯ i−¯yi}+τmax{0,¯yi−θ⊤¯ i}.
6Chap e 1. In oduc ion
This linea op imiza ion p oblem penalizes he ( aining) e o s ei=θ⊤¯ i−¯yi,i=
1, . . . , N in an asymme ic way. The posi i e e o s a e weigh ed wi h coe icien
(1 −τ) and he nega i es wi h coe icien τ. I τ∈(0,1) is close o ze o, hen he
posi i e e o s will be highly penalized (in compa ison wi h he nega i e ones).
This means ha e e y op imal solu ion θτ o he linea op imiza ion p oblem will
end o make mos o he e o s nega i e. This implies ha θ⊤
τ kcould be used as
a p obabilis ic lowe bound o yk. In a simila way, a p obabilis ic uppe bound
could be ob ained aking τ∈(0,1) close o 1. Unde a he mild assump ions,
any minimize θτo he p oposed op imiza ion p oblem can be used o ob ain
an es ima ion o he τquan ile. Tha is, θ⊤
τ kse es as an es ima ion o he τ
quan ile associa ed wi h yk. See [65], [71] and [66] o u he de ails.
One o he main limi a ions o quan ile eg ession is ha a la ge numbe o aining
samples Nis equi ed i one desi es o ob ain p obabilis ic gua an ees o he
me hod when τis chosen close o he ex emes o he in e al (0,1). This is due
o he ac ha es ima ing he p obabili y o a e e en s equi es a la ge numbe o
samples. Fo example, he numbe o independen iden ically dis ibu ed samples
equi ed o ob ain he 1 −ϵquan ile o a mono a iable andom a iable g ows
wi h 1
ϵ(see [72], [61] and [62]).
Compa a i ely, compu ing mul i a ia e p edic ion egions becomes a ha de ask.
The simples way would be o ob ain in e als conside ing each a iable inde-
penden ly and hen cons uc box-shaped egions (see sec ion 2.2.3 in [73]). The
main ad an age o his me hod lies in i s simplici y. Howe e , he desi ed p oba-
bili y may no be a ained o he size o he egions may be oo la ge. Boo s ap
me hods a e used in [74, 75] o cons uc egions ha con ain a pa h o a andom
a iable wi h a leas a ce ain p obabili y. This means ha he ob ained egions
a e ac ually in e als o a p−s ep p edic ion ins ead o a mul i a ia e sys em. In
a simila manne , [76, 75] calcula e p edic ion egions o Vec o Au o Reg ession
(VAR) models in he ield o econome ics. Following simila me hodologies, [77]
p o ides in e als o an en i e pa h o o ecas s in Ma ko p ocesses. The p oblem
o e ining p e iously es ima ed p edic ion egions in o de o a ain a co e age
p obabili y close o he desi ed one is esea ched in [78] and [79]. Howe e , all
hese a o emen ioned echniques do no deal wi h mul i a ia e ou pu sys ems.
Ins ead, he compu ed egions co esponds o he p edic ion o he same a iable
o di e en ime s eps.
In he ield o mul i a ia e ou pu sys ems, he li e a u e is a he sca ce. Fo ex-
ample, in [80], p edic ion egions o a simple mul i a ia e linea eg ession model
a e ob ained. On he o he hand, [81] p oposes a me hod o calcula e p edic ion
egions o a sys em by compu ing he Jacobian o he Pa ial Leas -Squa es Re-
g ession (PLS) pa ame e s. Thus, by means o his local linea iza ion, an ellipsoid-
shaped egion can be ob ained. Also, [82] manages o ob ain ellipsoidal egions
o dynamical sys ems by means o an In e se Reg ession (IR) scheme. This IR
scheme is mo e e icien and eliable han he classic eg ession app oaches when
1.1. Mo i a ion and objec i es 7
high dimensional da a is a ailable. In [83], a da a-d i en amewo k o gene a e
and e alua e ellipsoidal p edic ion egions o cha ac e ize he unce ain y o a
ime-se ies is p oposed. This me hodology is applied o he elec ici y p ices as a
pa h o ecas ing p oblem. In he ield o machine lea ning, Con o mal P edic ion
echniques [84] a e used o ob ain p edic ion egions o any me hod p oducing
a cen al p edic ion o he ou pu s. Simila ly, i is p oposed in [85] a amewo k
o ob ain a s ochas ic model based on a de e minis ic model o a dynamic sys-
em in he ield o obo ics. On he o he hand, assuming ha any ini e se o
samples ollows a mul i a ia e no mal dis ibu ion, Gaussian p ocesses (GPs) [23]
can also be used. The main limi a ion o his app oach is ha i elies hea ily on
he knowledge o he i s wo momen s o he unde lying mul i a ia e p obabili y
dis ibu ion.
1.1.3 Model p edic i e con ol
Once eached his poin , i is impo an o ecall ha he pu pose o inding a
good model o a dynamic sys em is because we wan o induce a ce ain beha iou
o hese sys ems, ha is, we wan o con ol hem. In his disse a ion, Model
P edic i e Con ol (MPC) [86, 87] has been chosen as he con ol algo i hm o be
used in he p oposed con ol examples.
MPC co esponds o a se o compu e con ol echniques sha ing some common
ideas. I s de elopmen exploded a he end o he 1970s hanks o he wo ks o
[88, 89], ecei ing a en ion om bo h academia and indus y. In hese wo ks,
hey used p edic ions ob ained by means o ini e impulse esponse models o
unca ed s ep esponse models o ob ain op imal con ol ac ions while minimizing
he acking e o o he ou pu o he sys em. Howe e , mos o hese algo i hms
had an heu is ic na u e, hey we e no able o deal wi h dis u bances and lacked
s abili y gua an ees.
This i s gene a ion o p edic i e con olle s was ollowed by he Gene alized
P edic i e Con ol (GPC) algo i hm [90, 91] which comp ised hese p e ious con-
olle s based on inpu -ou pu models. On he o he hand, he s a e-space in e -
p e a ion o MPC [92, 93] appea ed, and quickly became he s anda d o mula ion
in MPC. A good e iew o some p ope ies o he MPC con olle s, including s a-
bili y can be ound in [94].
The main ideas a ound MPC a e:
•MPC is an op imal con olle in he sense ha he inpu is calcula ed such
ha i minimizes a ce ain cos unc ion, mainly penalizing he acking
e o wi h espec o he desi ed e e ence and he con ol e o . This e o
can be compu ed hanks o a ma hema ical model o he sys em, which leads
us o he ollowing poin .
•As he name sugges s, an accu a e ma hema ical model desc ibing he be-
ha iou o he sys em is needed. I is possible o use whe he inpu -ou pu
8Chap e 1. In oduc ion
models o s a e-space models. Because o he nume ical na u e o he con-
olle (i.e. an op imiza ion p oblem is sol ed o compu e he alue o he
inpu ins ead o ob aining an explici con ol law), disc e e- ime models a e
usually used. Fu he mo e, as i is easie o gua an ee he s abili y o he
con olle using s a e-space models, hese a e also widely used.
•Ano he in e es ing poin o he MPC con olle s is hei abili y o ackle
cons ained con ol p oblems easily. As he inpu is ob ained by sol ing
an op imiza ion p oblem, i is e y na u al o add cons ain s. These con-
s ain s may e lec limi ed capabili y in he ac ua o s, some physical limi s
o he sys em o e en ackle secu i y conside a ions.
•The las idea would be he eceding ho izon scheme. This means ha e en
hough he alue o he inpu is compu ed o many u u e ime ins an s,
only he i s one ( ha is, he one co esponding o he ac ual ime ins an )
is applied, disca ding he es . Then, a he nex sampling s ep, a new whole
se o inpu s is compu ed aking in o accoun he new a ailable in o ma ion.
No e ha , i we applied he whole se o alues compu ed a he i s ime
ins an , his would be open-loop con ol. Thus, he eceding ho izon scheme
p o ides eedback o he con olle .
F om hese p ope ies o he MPC con olle s, i is easy o see he many ad an ages
ha hey p esen , i.e.
•The con ol p oblem is o mula ed in he ime domain, in a lexible and
in ui i e manne in con as o o he con ol o mula ions ha need o be
designed in he equency domain. Also, i is applicable o any sys em
wi hou ega d o i s open-loop s abili y, delays, e c.
•In he mos gene al case, i allows o conside linea and nonlinea , uni a ia e
and mul i a ia e sys ems alike using he same o mula ion o he con olle .
•I p esen s delay compensa ion explici ly and measu able dis u bances can
also be easily compensa ed.
•I is possible o ake in o accoun he knowledge o he e olu ion o he
e e ence in o de o ack he signal o he e e ence be o e i changes.
•I is possible o deal easily wi h cons ain s.
Howe e , he e a e also some d awbacks
•A su icien ly p ecise model o he sys em o be con olled is needed. Tha
is, he pe o mance o he con olle depends hea ily on he quali y o he
model. An inapp op ia e model o he sys em can lead o undesi ed pe o -
mance o he con olle in closed-loop ope a ion.
•I is necessa y o sol e an op imiza ion p oblem a e e y ime ins an k.
E en hough nowadays he compu a ional powe o he compu e s ha e
1.1. Mo i a ion and objec i es 9
been inc easing eno mously, many nonlinea MPC (NMPC) o obus MPC
p oblems s ill emain in ac able in p ac ice due o he high compu a ional
cos o he algo i hms.
As i was discussed p e iously, i is e y common ha he dynamics o he p ocess
o con ol a e almos comple ely unknown, o cing us o ely on black-box models.
Anyway, in mos o hese cases, he e exis s an app op ia e model s uc u e in
he li e a u e ha is able o e lec he dynamics o he sys em and hus inding
a good model is no an impossible ask.
On he o he hand, wi h espec o he compu a ional cos , he e a e se e al
ways o add ess hese p oblems, like de eloping speci ic op imiza ion algo i hms
o sol e MPC p oblems [95, 96] o eso ing o sub-op imal MPC schemes [97, 98].
Ano he way would be o ely on explici MPC implemen a ions [99, 100]. He e,
an analy ical solu ion o he MPC con olle can be ob ained i he op imiza ion
p oblem can be cas ed as a ce ain class o quad a ic p og amming (QP) p ob-
lems [101, 102]. The ob ained con ol law is piece-wise linea , ha is, di e en
linea s a e eedback con ol policies a e ob ained o di e en polyhed al egions.
Howe e , he numbe o egions g ows up la gely wi h espec o he numbe o
s a es, cons ain s, e c. becoming in ac able o many MPC p oblems.
Besides he a o emen ioned classic MPC schemes, a huge numbe o MPC a ian s
ha e been esea ched o e ime, i.e. app oaches which conside s dis u bances and
model misma ches. Ac ually, he classic de e minis ic MPC app oach p esen s
some inhe en obus ness [103, 104], which can be s udied by means o he Inpu -
o-S a e s abili y (ISS) [105] o obus ly asymp o ically s abili y [106] schemes.
Howe e , i is no possible o gua an ee ha he cons ain s a e ul illed in a
obus manne . Fo ha pu pose, a desc ip ion o he unce ain ies is necessa y.
This led o min-max MPC, whe e he wo s -case alues o he dis u bances we e
aken in o accoun o compu e he con ol ac ions [107, 108]. One s ep u he
was o op imize eedback con ol laws ins ead o con ol ac ions o imp o e he
obus ness o he con olle [109]. The main d awback o his subclass o MPC
con olle s is ha hei online compu a ional ime is inc edibly high, some imes
becoming in ac able, especially when op imizing eedback con ol laws.
Since hen, nume ous obus MPC schemes ha e been p oposed. The mos impo -
an one is p obably he ube-based MPC [110, 111, 112]. He e, a nominal con ol
ac ion is compu ed alongside a p opo ional policy. As he policy is assumed o be
a p opo ional gain, he op imiza ion p oblem is no in ini e dimensional and hus
i is gene ally ac able. Also, one could ely on cons ain igh ening schemes [27]
o ul ill he cons ain s unde he p esence o any kind o dis u bance.
Howe e , obus MPC is, in gene al, e y conse a i e because i add esses al-
ways he wo s case, ha is, i is always conside ing he ex eme alues o he
dis u bances. Fo ha eason, s ochas ic MPC [113, 114] and scena io app oaches
[115, 116] which conside s he p obabili y dis ibu ion o he dis u bances ha e
10 Chap e 1. In oduc ion
a ac ed he a en ion o many esea che s. These echniques allow us o de elop
con olle s ha ul ill he cons ain s wi h a speci ied p obabili y, i.e. elaxing he
wo s case op imiza ion o obus MPC. Fo example, in he scena io app oach,
his can be done by aking a ini e numbe o ealiza ions o he dis u bance du -
ing he online compu a ion o he con ol ac ion. In his app oach, howe e , he
numbe o scena ios o be gene a ed inc eases wi h he dimension o he p oblem,
leading o una o dable compu a ional imes in many cases.
As mos o hese app oaches complica e he online op imiza ion p oblem o be
sol ed online, one could wonde i i is possible o ob ain a s ochas ic MPC con-
olle while keeping he online op imiza ion p oblem as simple as he de e minis-
ic MPC p oblem. This leads us o p obabilis ic alida ion app oaches [117, 118]
whe e, by means o o line simula ions o he closed-loop sys em, i is possible o
de e mine i he con olle ul ills some p obabilis ic speci ica ions. This means
ha he compu a ional bu den is ou side he con ol loop and hus he online
op imiza ion p oblem can be sol ed easily. Also, i can be applied o any con-
olle , e en i i is no an MPC con olle . Howe e , his me hod only p o ides a
“yes” o “no” answe wi hou assessing nume ically he pe o mance o he con-
olle , which makes i ha d o compa e wi h o he con olle s. In some cases,
i could e en lead o non easible solu ions i none o he con olle s sa is ies he
p obabilis ic cons ain .
O he popula a ian s o he s anda d MPC con olle a e he ollowing:
•Economic MPC [119, 120] allows o op imize di e en cos unc ions in o de
o imp o e he p o i abili y o a ce ain p ocess. The s abili y esul s o
s anda d MPC a e no , in gene al, applicable o economic MPC. Howe e ,
unde some assump ions [121], i was p o en ha i is possible o ind a
Lyapuno unc ion o he closed-loop sys em in o de o ensu e s abili y.
•Dis ibu ed MPC [122, 123] conside s a decen alized con ol sys em, educ-
ing he compu a ional bu den o he indi idual op imiza ion p oblems and
becoming mo e scalable and obus o ailu es, which may be help ul when
con olling la ge-scale sys ems. Howe e , he pe o mance may be wo se
han he pe o mance o he cen alized con olle .
•MPC o acking [124, 125] ackles he p oblem o a bi a ily changing he
e e ence signal o he con olle . This app oach p ese es s abili y and
gua an ees ha he sys em can be s ee ed o any admissible equilib ium
poin , no ma e wha he ini ial equilib ium poin is, wi hou losing he
ecu si e easibili y. This s a egy also p o ides a la ge domain o a ac ion
o a gi en p edic ion ho izon and, in he case o un eachable e e ences,
asymp o ic con e gence o he closes eachable e e ence is p o en as well,
only a he expense o adding a ew mo e decision a iables.
1.2. Ou line 11
1.1.4 Objec i es o his disse a ion
Many di e en esea ch ields ha e been p esen ed h oughou his in oduc o y
chap e : sys em iden i ica ion, in e al p edic ions, MPC, e c. This disse a ion
is o ien ed owa ds de eloping new me hods ha may imp o e he pe o mance
o he s a e-o - he-a me hods in each one o he p e iously p esen ed ields.
Summa izing, he objec i es o his disse a ion a e:
1. De eloping new o ecas ing schemes o ime-se ies and nonlinea sys ems.
I is in e es ing o ob ain no only he expec ed ou pu bu also a egion
whe e he ou pu can be ound wi h a ce ain p obabili y.
2. P oposing new da a-d i en black-box models o be used in MPC s a egies.
As he p oposed models may e lec be e he beha iou o ce ain sys ems,
hey migh inc ease he pe o mance o ce ain con olle s.
3. Ob aining p obabilis ic bounds on he cons ain iola ion le el o any kind
o con olle in a s ochas ic se ing. These bounds could be used o une he
hype -pa ame e s o a con olle in a so -cons ain app oach.
Taking in o accoun hese objec i es, he ollowing esul s we e ob ained:
1. Co esponding o he i s objec i e, some p edic o s based on dissimila i y
unc ions a e p oposed. These include nominal p edic ions, in e al p edic-
ions and p edic ion egions. All hese me hods we e es ed agains some
baseline echniques in many examples and showed a be e pe o mance.
2. Fo he second objec i e, i is shown ha , by means o he a o emen ioned
dissimila i y unc ion, i is possible o iden i y a model om he a ailable
inpu -ou pu da a. Th ough di e en nume ical examples, i can be seen
ha he model pe o ms be e han o he machine lea ning echniques.
The model has been used wi hin a MPC o acking amewo k o show he
e ec i eness o he app oach in con ol p oblems.
3. Finally, wo di e en sha p bounds on he cons ain iola ion a e we e
ob ained. These bounds we e used in he con ex o da a cen e ene gy
op imiza ion o quan i y he quali y o se ice p o ided o he use s, p o ing
i s use ulness.
1.2 Ou line
All he esul s o his disse a ion con e ge o MPC a some poin , ha is, hey
a e use ul o imp o e he pe o mance o MPC con olle s o hey allow o de elop
new MPC schemes ha may a ain be e esul s han o he s a e-o - he-a MPC
con olle s. Fo he sake o simplici y, he con ibu ions a e di ided in o h ee
g oups: no el algo i hms o p edic ion and o ecas ing, new modeling me hods
12 Chap e 1. In oduc ion
o con olle design and, inally, andomized algo i hms o s ochas ic con ol.
Ha ing his idea in mind, he ex was di ided in o h ee di e en pa s.
Pa I p esen s he de eloped p edic o s based on dissimila i y unc ions. These
p oposed me hodologies a e ela ed o he ones appea ing in he con ex o Di ec
Weigh Op imiza ion [57] and he K iging me hod [126, 127]. Fi s , he concep
o dissimila i y unc ion is exposed, along wi h he p oposed dissimila i y unc ion
o his disse a ion. F om ha , a nominal p edic o can be easily ob ained. Gi en
a eg esso , a p edic ion is ob ained as a combina ion o pas ou pu s o he sys em
using some weigh s ha a e ob ained by means o an op imiza ion p oblem. A e
ha , a me hod o ackle he p oblem o unce ain y quan i ica ion is p oposed.
Again, by means o he a o emen ioned dissimila i y unc ions, i is shown ha
i is possible o ob ain in e al p edic o s o he uni a ia e case and p edic ion
egions o he mul i a ia e case.
Pa I is di ided in o wo chap e s:
Chap e 2 wo ks as an in oduc ion o chap e 3, in oducing he basics o dis-
simila i y unc ions and p oposing a i s p edic o o ob ain he expec ed alue o
a ime-se ies. A nume ical example o he o ecas ing o he Dow Jones Indus ial
A e age Index is shown o e i y he good pe o mance o he algo i hm.
Chap e 3 p esen s he me hodologies o quan i y he unce ain y o he p e-
dic ions. Fi s , a me hod o ob ain an empi ical p obabili y densi y unc ion o
a uni a ia e unc ion is discussed. F om his empi ical p obabili y dis ibu ion,
i is possible o compu e quan iles o a desi ed p obabili y, ob aining an in e al
p edic o . La e , he me hodology is ex ended o ackle he mul i a ia e case. As
ob aining he p obabili y dis ibu ion o a mul i a ia e andom a iable is gene -
ally in ac able, an al e na i e me hod bypassing he p obabili y densi y unc ion
allows us o ob ain p edic ion egions o such a iables. Some nume ical examples
a e p esen ed in o de o show he imp o emen s wi h espec o o he baseline
echniques.
On he o he hand, pa II p esen s no el me hodologies o ob ain models o
nonlinea sys ems o ime-se ies. He e, he me hodologies p oposed in pa I a e
ex ended in o de o ob ain a model o he sys em by eed backing he 1-s ep
ahead p edic ions and egula izing he op imiza ion p oblem used o compu e
such p edic ions. Two di e en models a e p oposed in his chap e : a linea
ime a ying (LTV) model ob ained by weigh ing app op ia ely he local da a
wi hin he da a se and a ke nel-based e sion o he s a e-space k iging whe e
he non-linea i y is modeled by means o a ke nel unc ion. The ob ained models
can be used easily o make p edic ions o design MPC con olle s. Fo ecas ing
and Con ol examples a e bo h p o ided o show he e ec i eness o he p oposed
app oaches.
Pa II is di ided in o wo chap e s:
1.3. Publica ions 13
Chap e 4 p esen s he s a e-space k iging app oach, named a e he k iging
me hod due o i s simila i ies and also because he weigh s used o ob ain he
p edic ions become he s a e o he new model. This chap e is ocused on au-
onomous sys ems and ime-se ies. Thus, i p esen s me hodologies o ob ain a
model o he ou pu s om pas da a o he sys em ha can be used o make
p edic ions. Nume ical examples wi h compa isons a e p o ided in o de o show
he e ec i eness o he p oposed app oaches.
Chap e 5 in oduces he s a e-space k iging o non-au onomous sys ems. I is
shown ha he inpu can be conside ed easily wi hin he op imiza ion p oblem
wi h e y ew changes. As he app oach conside ed in his chap e can ackle he
p oblem o o ecas ing sys ems wi h manipulable inpu s, he p oposed app oach
is sui able o design, o example, MPC con olle s whose p edic ion model is a
s a e-space k iging model. The s abili y o he p oposed con olle is p o en and a
nume ical example and a eal expe imen a e p o ided o e i y he pe o mance
o he con olle .
Finally, pa III p esen s no el bounds on he cons ain iola ion a e ha
can be used o quan i y he pe o mance o di e en con olle s in a s ochas ic
se ing in o de o choose he mos app op ia e con olle among hem. These
con olle s a e no necessa ily MPC con olle s, ha is, any con olle can be
assessed by means o hese echniques. Ano he ad an age o hese me hods is
ha he compu a ional bu den o he p ocess is comple ely o line and hus he
online compu a ion ime o he con olle does no change. The p oposed me hods
a e inally alida ed in he con ex o he managemen o a da a cen e .
Pa III is di ided in o wo chap e s:
Chap e 6 p esen s he concep s needed o ob ain he bounds on he empi ical
cons ain iola ion le el based on he esul s by Che no in [128] and by Alamo
e al. in [61]. These esul s p o ides wo di e en exp essions ha can be used o
measu e he eliabili y o any con olle .
Chap e 7 in oduces he model o a da a cen e (dealing wi h bo h he he mal
modeling and he ask model o he compu e s) ha will be used as a benchma k
o he p e iously de eloped bounds. In his chap e , no only he model o he
da a cen e and he implied cons ain s a e p esen ed, bu a managemen app oach
based on MPC is also p oposed. As he p oblem o his speci ic sys em is gene ally
ha d o sol e (in ege op imiza ion, non-con ex model, e c.) he solu ion o he
p oblem is compu ed by means o a pa icle-based sol e implemen ed in a G aphic
P ocessing Uni (GPU).
1.3 Publica ions
The esul s shown h oughou his hesis ha e been published in se e al jou nals
and con e ence p oceedings, al hough some o hem a e s ill unde e iew.
20 Chap e 2. Fo ecas ing using dissimila i y unc ions
coo dina e sys em. Howe e , many o he dissimila i y unc ions ha can be
ound in he li e a u e a e no in a ian . Fo example, any dissimila i y unc ion
based on he dis ance o d o he elemen s o D, such as ha o equa ion (2.1),
will be dependen on he pa icula choice o he coo dina e sys em.
The p oposed op imiza ion p oblem (2.2) is a s ic con ex op imiza ion p oblem
subjec o con ex cons ain s. This means ha i has a unique solu ion [139].
No e ha he nume ical esolu ion can be add essed using a dual o mula ion. In
he dual o mula ion o his pa icula op imiza ion p oblem, he numbe o dual
decision a iables is equal o he numbe o equali y cons ain s (n+ 1) which is
in many si ua ions much smalle han he numbe o p imal a iables (N). On
he o he hand, he g adien o he objec i e unc ion in he dual o mula ion
can be ob ained in a di ec way because once he dual a iables a e ixed, he
op imal alues o he p imal a iables a e ob ained by sol ing Nsepa able op i-
miza ion p oblems. The compu a ions in his disse a ion ha e been made using
an accele a ed g adien me hod in he dual a iables (see [140], [141] and [142]).
As i is o mally s a ed in he ollowing p ope y, he op imiza ion p oblem has
an explici solu ion o he pa icula case γ= 0.
P ope y 2.7. J0(d, D) has he ollowing explici exp ession
J0(d, D) = N−1+ (d−dc)⊤(DD⊤−Ndcd⊤
c)−1(d−dc),
whe e dc=N−1D1⊤and 1⊤is a column ec o wi h all i s Ncomponen s equal
o 1.
P oo . The op imiza ion p oblem is sol ed using a dual o mula ion whe e µ∈Rn
deno es he mul iplie s associa ed wi h he equali y cons ain
d=
N
X
i=1
λ(i)di=Dλ,
and νis he mul iplie co esponding o he equali y
1 =
N
X
i=1
λ(i)=1λ.
he Lag ange unc ion is
L(λ, µ, ν) = λ⊤λ+µ⊤(Dλ −d) + ν(1λ−1).
Deno e λ∗,µ∗and ν∗ he op imal alues o he p imal and dual a iables. F om
∂L(λ∗,µ∗,ν∗)
∂λ = 0, i is clea ha he op imal ec o λ∗is gi en by
λ∗=−1
2(D⊤µ∗+1⊤ν∗).(2.3)
2.1. P oposed dissimila i y unc ion 21
Since 1λ∗= 1, D1⊤=Ndcand 11⊤=N, i is possible o p emul iply bo h e ms
o he las equali y by 1 o ob ain
1 = −1
2(1D⊤µ∗+Nν∗)
=−N
2(d⊤
cµ∗+ν∗).
The e o e,
ν∗=−2
N−d⊤
cµ∗.
Subs i u ing he exp ession o ν∗in (2.3) yields
λ∗=−1
2D⊤µ∗−1⊤(2
N+d⊤
cµ∗)
=1⊤
N−1
2(D⊤−1⊤d⊤
c)µ∗.(2.4)
P emul iplying by D, he ollowing equa ion is ob ained
Dλ∗=dc−1
2(DD⊤−Ndcd⊤
c)µ∗.(2.5)
F om he equali y cons ain Dλ∗=dand (2.5), he alue o µ∗can be ob ained
as
µ∗=−2(DD⊤−Ndcd⊤
c)−1(d−dc).
Subs i u ing µ∗in equa ion (2.4), i is possible o compu e λ∗as
λ∗=1⊤
N+ (D⊤−1⊤d⊤
c)(DD⊤−Ndcd⊤
c)−1(d−dc).
Finally, aking in o accoun ha
1(D⊤−1⊤d⊤
c) = (Nd⊤
c−Nd⊤
c)=0,
he ollowing exp ession is ob ained
(D⊤−1⊤d⊤
c)⊤(D⊤−1⊤d⊤
c) = DD⊤−Ndcd⊤
c.
F om las equali y and he exp ession o λ∗, he alue o J0(d, D) can be compu ed
as
J0(d, D)=(λ∗)⊤λ∗=N−1+ (d−dc)⊤(DD⊤−Ndcd⊤
c)−1(d−dc).■
The p e ious esul shows ha he dissimila i y unc ion is a quad a ic unc ion
on he a gumen d o he pa icula case γ= 0. Fo he mo e gene al case
in which γ > 0, i is possible o in e om he Ka ush-Kuhn-Tucke (KKT)
op imali y condi ions [139] ha he dissimila i y unc ion Jγ(d, D) is a piecewise
con ex quad a ic unc ion wi h espec o d.
22 Chap e 2. Fo ecas ing using dissimila i y unc ions
2.1.1 Cla i ying example
Figu e 2.1 shows a cloud o poin s ob ained om a ce ain p obabili y dis ibu ion.
On he le , he o iginal cloud o poin s is shown and, on he igh , his cloud o
poin s is o a ed an angle π/4.
Red ma ke s co espond o he poin s in he da a se Dwhe eas blue ma ke s
co espond o wo gi en alues o d, i.e.
d1=0.25
1, d2=0.75
1.
No e ha hese alues co espond o he le igu e, whe e hey a e no o a ed.
Assuming ha γ= 0, he ob ained alues o he dissimila i y o hese ec o s d1
and d2a e
J0(d1, D)=0.0056, J0(d2, D)=0.0191.
I is easy o see ha he dissimila i y is smalle o d1because he concen a ion
o da a poin s is la ge in ha egion, in con as o d2whe e he concen a ion
o ed ma ke s is small. This means ha d1is a mo e likely e en han d2.
-0.5 0 0.5 1 1.5
-0.5
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Figu e 2.1: Le : o iginal cloud o poin s. Righ : o a ed cloud o poin s.
Also, no e ha , as i was p o en in p ope y 2.6, he alue o he dissimila i y
unc ion is in a ian wi h espec o a ine ans o ma ions.
2.2 Dissimila i y unc ions and eg ession
This sec ion shows how dissimila i y unc ions can be used in he con ex o e-
g ession. Deno ing he eg esso as z∈Rnzand he ou pu as y∈Rny, i is
assumed ha he ollowing da a se s Zand Ya e a ailable
Z=¯z1¯z2. . . ¯zN, Y =¯y1¯y2. . . ¯yN,
whe e ¯yi∈Rnyand ¯zi∈Rnz∀i= 1, ..., N a e pas samples o he ou pu s and
he eg esso espec i ely.
2.2. Dissimila i y unc ions and eg ession 23
Fixing a ce ain zand compu ing Jγ z
y,Z
Ygi es us he dissimila i y
o a ce ain ou pu ycondi ioned o he alue o he eg esso z. Thus, he
dissimila i y unc ion becomes
Jγ z
y,Z
Y= min
λ(1),...,λ(N)
(1 −γ)
N
X
i=1
λ2
(i)+γ
N
X
i=1 |λ(i)|(2.6a)
s. . z=
N
X
i=1
λ(i)¯zi(2.6b)
y=
N
X
i=1
λ(i)¯yi(2.6c)
1 =
N
X
i=1
λ(i).(2.6d)
Gi en z, conside , om all he possible alues o y, he one ha minimizes he
dissimila i y unc ion de ined by he p e ious op imiza ion p oblem
˜y(z) = a g min
yJγ z
y,Z
Y.(2.7)
No e ha minimizing he dissimila i y co esponds o maximizing he simila i y.
Thus, ˜y(z) is he mos likely alue o happen gi en he eg esso zand he da a
se s Zand Y. This ˜y(z) could be used as a o ecas ing o he nex ou pu .
Now, ins ead o Jγ z
y,Z
Y, conside Jγ(z, Z), ha is,
Jγ(z, Z) = min
λ(1),...,λ(N)
(1 −γ)
N
X
i=1
λ2
(i)+γ
N
X
i=1 |λ(i)|(2.8a)
s. . z=
N
X
i=1
λ(i)¯zi(2.8b)
1 =
N
X
i=1
λ(i).(2.8c)
As he op imiza ion p oblem in (2.8) is a less cons ained e sion o ha in (2.6),
he op imal alue o he cos o he mo e cons ained p oblem is always g ea e
o equal o he op imal alue o he cos o he less cons ained one, ha is,
Jγ(z, Z)≤Jγ z
y,Z
Y.
24 Chap e 2. Fo ecas ing using dissimila i y unc ions
Now, deno e he ec o o weigh s λ ha minimizes he op imiza ion p oblem in
(2.8) as λ∗. F om his op imum ec o o weigh s λ∗, ˜y(z) can be compu ed as
˜y(z) = Y λ∗=
N
X
i=1
λ∗
(i)¯yi.(2.9)
Then, i is easy o see ha
Jγ(z, Z) = Jγ z
˜y(z),Z
Y
and hus ˜y(z), compu ed using he op imal weigh s om (2.8), is he one ha
minimizes he dissimila i y unc ion o a ixed z(see equa ion (2.7)). No e ha
his o ecas ing is made by means o a linea combina ion o pas ou pu s whose
weigh s a e compu ed ob aining an a ine en elope o he eg esso z. As i is
explained in wha ollows, s ic ly posi i e alues o γencou age he componen s
o λ o be posi i e (which means ha he cen al es ima ion is o en ob ained
om an in e pola ion o poin s).
I is easy o see ha when e e y λ(i)≥0,∀i= 1, ..., N, hen PN
i=1 |λ(i)|= 1.
Howe e , when some o he componen s o λa e nega i e, PN
i=1 |λ(i)|>1. In
o he wo ds, he cos e m PN
i=1 |λ(i)|becomes la ge when ex apola ing poin s
(i.e. using nega i e alues o λ(i)). This means ha con ex combina ions o
λ(i)a e encou aged and hus in e pola ion is p e e ed, which may imp o e he
p edic ions when nonlinea sys ems a e aken in o accoun . The op imal alue
o γ≥0 could be ob ained in di e en ways. A easonable choice could be,
o example, o choose he one ha minimizes he e o o he p edic ions wi h
espec o he eal ou pu s in a alida ion se .
As i is s a ed in he ollowing p ope y, he es ima ion ob ained o he pa icula
case γ= 0 ma ches he one gi en by he linea leas squa es me hod.
P ope y 2.8. Gi en z, he es ima ion
˜y(z) = a g min
yJ0 z
y,Z
Y,
ma ches he es ima ion ob ained by linea leas squa es.
P oo . F om equa ion (2.9), he op imal alue o he es ima ion is ˜y(z) = ΣN
i=1λ∗
(i)¯yi,
whe e o he pa icula case γ= 0, λ∗
(i),i= 1, . . . , N, a e he op imal alues o
2.2. Dissimila i y unc ions and eg ession 25
he op imiza ion p oblem
min
λ(1),...,λ(N)
N
X
i=1
λ2
(i)
s. . z=
N
X
i=1
λ(i)¯zi
1 =
N
X
i=1
λ(i).
De ining
R=¯z1¯z2. . . ¯zN
1 1 . . . 1, =z
1,
he equali y cons ain s can be ew i en as
Rλ = . (2.10)
F om he KKT op imali y condi ions, he op imal solu ion is gi en by (see sub-
sec ion 10.1.1 in [139])
INR⊤
R0(nz+1) λ∗
φ∗=0
,
whe e φ∗co esponds o he op imal dual decision a iables co esponding o he
equali y cons ain (2.10). The p e ious equa ion can be ew i en as
λ∗=−R⊤φ∗
Rλ∗= .
F om he e, i is in e ed ha −RR⊤φ∗= , which implies φ∗=−(RR⊤)−1 and
hus inally ob aining
λ∗=R⊤(RR⊤)−1 .
The e o e,
˜y(z) = Y⊤λ∗=Y⊤R⊤(RR⊤)−1 = ⊤(RR⊤)−1RY.
This co esponds o he leas squa es es ima ion ob ained when he conside ed
eg esso s a e he ec o s ¯z⊤
i1⊤,i= 1, . . . , N (see [143], [64]). ■
P e ious p ope y shows ha he p oposed es ima ion me hod encompasses he
leas squa es me hod o he pa icula case γ= 0. A amily o op imal es ima o s
is ob ained i γis conside ed a uning pa ame e .
F om now on, in o de o make he manusc ip mo e eadable, he no a ion will
be simpli ied by emo ing Zand Y om he dissimila i y unc ion (since hey a e
assumed o be ixed). Tha is, he no a ion becomes
Jγ z
y,Z
Y=Jγ(z, y).
26 Chap e 2. Fo ecas ing using dissimila i y unc ions
Mo eo e , he minimal alue o he dissimila i y unc ion gi en zwill be deno ed
as J∗
γ(z), ha is
J∗
γ(z) = Jγ(z, ˜y(z)) = min
yJγ(z, y).
The ollowing ema k summa izes he p ocess o use he dissimila i y unc ions in
he con ex o eg ession.
Rema k 2.9. Gi en he eg esso z, he dissimila i y unc ion Jγ(·,·)and he
da a se s Zand Y, sol e op imiza ion p oblem in (2.8) o ob ain λ∗. Once λ∗has
been ob ained, he o ecas ing ˜y(z)is compu ed by means o equa ion (2.9).
2.3 Applica ion: o ecas ing s ock p ices using dissim-
ila i y unc ions
In his sec ion, a nume ical example is p esen ed o show he e ec i eness o he
p oposed dissimila i y unc ion as a p edic o .
Conside he e olu ion o he p ice o a s ock as a ime se ies pk∈ P, whe e k
is a ime uni and Pis he ange o alues ha he s ock p ice can ake. The
s a e o his ime se ies can be desc ibed as he alue a ime ko a se ies o
echnical indica o s. These echnical indica o s can be pas alues o he p ice,
s ock p ice e u ns o mo e complex me ics in gene al. Thus, his g oup o
echnical indica o s will o m he eg esso zk. The objec i e is o be able o
p edic up o l-s eps (i.e. l-days) ahead he p ice o a s ock, ha is, o ob ain
˜pk+la ime k. Thus, ykco esponds o
yk=pk+1 pk+2 . . . pk+l⊤.
As i was shown in he p e ious sec ion, Zand Ya e compounded o pas samples
o zand y, deno ed as ¯ziand ¯yi.
The app oach conside ed in his example uses only a small subse o he da abase,
deno ed as Ω(zk), o compu e he p edic ions ˜pk+l. This subse Ω(zk) is chosen
by inding he NΩcloses poin s o zkwi hin he da a se by means o a ce ain
dis ance measu emen . Thus, Ω(zk) is compounded o NΩpai s (¯zi,¯yi) o he ull
da a se .
Once he da a o be included in Ω(zk) a e selec ed, he op imiza ion p oblem (2.8)
is sol ed wi h his educed da a se . Also, no e ha he o ecas ing o pk+lis also
made wi h his small po ion o he comple e da a se . Algo i hm 4 gi es a o mal
desc ip ion o he p oposed app oach.
Rema k 2.10. The dis ance can be any measu e o how close a e zk o ¯zi. Eu-
clidean dis ance would be he mos ypical choice, bu i is also possible o conside
o he aspec s like he ime span be ween samples. In his way, ecen da a could
2.3. Applica ion: o ecas ing s ock p ices using dissimila i y unc ions 27
Algo i hm 1: l-s ep ahead s ock o ecas ing using local da a
Da a: Z,Y,zk,NΩand γ.
Resul : ˜pk+1 ˜pk+2 . . . ˜pk+l.
1Compu e he dis ance o each ¯ziin he da abase o zk;
2C ea e a lis o he en ies in he da a base so ed acco ding o he compu ed
dis ances. Deno e ¯zjand ¯yjas he eg esso ¯ziand he ou pu ¯yio he j- h
en y in his o de ed lis espec i ely;
3Build Ω(zk) using he i s NΩen ies o he o de ed lis , ha is,
Ω(zk)≜{(¯zj,¯yj)} ∀j= 1, . . . , NΩ.
4Sol e he ollowing op imiza ion p oblem:
min
λ(1), ..., λ(NΩ)
(1 −γ)
NΩ
X
j=1
λ2
(j)+γ
NΩ
X
j=1 |λ(j)|
s. .
NΩ
X
j=1
λ(j)= 1,
NΩ
X
j=1
λ(j)¯zj=zk.
and compu e he o ecas ed p ices as:
˜pk+1 ˜pk+2 . . . ˜pk+l⊤= ˜yk=
NΩ
X
j=1
λ(j)¯yj.
be p io i ized when selec ing he elemen s o Ω(zk). O he aspec s like seasonali y
could also be aken in o accoun .
2.3.1 Resul s
The da a se o be used in his example was ob ained om he da a p o ide
Bloombe g and i is composed o he daily closing p ices o he Dow Jones Index
om 2005 o 2016. The da a was di ided in o a aining se (Zand Y) om 2005
o 2014 and a es se om 2015 o mid-2016. This pe iod was chosen because
he ma ke does no ollow a ce ain end (bullish o bea ish) ha would make
he o ecas ing i ial. In o de o educe he noise o he ime se ies, he p ices
a e smoo hed using a 5-day Exponen ial Mo ing A e age (EMA) compu ed as:
pm
EMA,k =2
m+ 1pk+1−2
m+ 1pm
EMA,k−1,
28 Chap e 2. Fo ecas ing using dissimila i y unc ions
wi h pm
EMA,0=p0and m= 5 in his case. No e ha he smoo hing applied he e
is e y ligh in compa ison o he usual alues o mused in sho - e m o ecas ing
(12 and 26 day EMA [144]). This p ese es as p ice luc ua ions a he cos o
making he o ecas ing p ocess ha de .
The eg esso zkis composed o he las en days smoo hed p ices, as well as he
5-day and 10-day ela i e di e ence pe cen age o unsmoo hed p ices (RDP) [145]
i.e.
RDPm
k= 100 pk−pk−m
pk,
being mequal o 5 and 10 espec i ely. Thus,
zk=pkpk−1. . . pk−9RDP5
kRDP10
k⊤.
The size o Ω(zk) is se o NΩ= 250 and γ= 0. The o ecas and eal p ices can
be seen in igu es 2.2 and 2.3. The esul s show ha he o ecas ing is accu a e
enough a i s and i becomes wo se as he p edic ion ho izon inc eases, as i can
be expec ed.
In o de o compa e he esul s ob ained wi h he p oposed app oach, he esul s
ob ained wi h a pe sis ence p edic o and a Neu al Ne wo k (NN) a e shown.
This pe sis ence p edic o wo ks as a ma ingale, ha is, ˜pk+l=pkwhe eas he
NN co esponds o a mul i-laye pe cep on wi h 20 neu ons in he hidden laye
and ained ollowing he Le enbe g-Ma qua d ule. The nume ical esul s a e
shown in able 2.1. In pa icula , he mean squa ed e o s (MSE) o he p oposed
app oach and he a o emen ioned baselines a e shown in able 2.1a. I can be
seen ha he p oposed app oach achie es he lowes e o s. Howe e , no e ha
o long s ep ahead p edic ions, he pe sis ence p edic o achie es esul s as good
as he o he echniques, a consequence om he long e m andom walk na u e
o he inancial ma ke .
Table 2.1: Resul s o he Dow Jones o ecas ing example.
(a) Mean squa ed e o s (MSE).
kP oposed MLP Pe sis ence
1 3,294.8 3,577.0 5,296.4
2 12,433.2 14,190.8 16,867.4
3 26,659.0 30,829.6 31,975.0
4 45,475.9 50,794.9 49,072.4
5 66,859.6 77,017.5 66,943.4
(b) S anda d de ia ion o he e o s (σ).
P oposed MLP Pe sis ence
57.4 59.9 72.8
111.5 119.3 130.0
163.1 175.6 178.9
212.9 224.9 221.6
257.8 276.0 258.8
On he o he hand, able 2.1b shows he s anda d de ia ion o bo h he p oposed
app oach and he baselines. Again, he p oposed app oach a ains he lowes
s anda d de ia ion in he e o s, which means ha he e o s a e no only smalle
in mean bu also hey a e mo e concen a ed.
2.3. Applica ion: o ecas ing s ock p ices using dissimila i y unc ions 29
0 50 100 150 200 250 300
Day
1.55
1.6
1.65
1.7
1.75
1.8
1.85
Closing P ice
104
Fo ecas
Real
0 50 100 150 200 250 300
Day
1.55
1.6
1.65
1.7
1.75
1.8
1.85
Closing P ice
104
Fo ecas
Real
0 50 100 150 200 250 300
Day
1.55
1.6
1.65
1.7
1.75
1.8
1.85
Closing P ice
104
Fo ecas
Real
Figu e 2.2: Fo ecas ed and eal p ices (5-day EMA) o 1 o 3 days o ecas ing.
36 Chap e 3. P obabilis ic p edic ion egions
Algo i hm 2: In e al es ima ion [y−
τ(z, γ, c), y+
τ(z, γ, c)].
Da a: z,τ∈(0,1), γ≥0, c≥0, Z,Y,ˇ
Y.
Resul : y−
τ,y+
τ.
1Ob ain he dissimila i y unc ion (see De ini ion 2.1) o each elemen o ˇ
Y:
aj=Jγ(z, ˇyj), j = 1, . . . , M.
2Compu e he condi ioned p obabili ies (see equa ion 3.1):
pj= ecpγ,c (z, ˇyj) = e−caj
M
P
ℓ=1
e−caℓ
, j = 1, . . . , M.
3Compu e he indexes ℓ+
τand ℓ−
τco esponding o he lowe and uppe
condi ioned τ-quan iles (see (3.2) and (3.3)):
ℓ+
τ= smalles in ege ℓsa is ying
ℓ
P
j=1
pj≥1−τ,
ℓ−
τ= la ges in ege ℓsa is ying
M
P
j=ℓ
pj≥1−τ.
4Make y−
τ= ˇyℓ−
τand y+
τ= ˇyℓ+
τ.
Conside now he ole o pa ame e c≥0 in he disc e e empi ical condi ioned
dis ibu ion gi en in equa ion (3.1). On he one hand, he choice c= 0 p o ides
a la dis ibu ion in which each elemen o ˇ
Yhas a condi ioned p obabili y equal
o 1
M. On he o he hand, la ge alues o cp o ide na ow dis ibu ions cen e ed
a ound he poin in ˇ
Y ha minimizes, gi en z, he dissimila i y unc ion Jγ(·,·),
i.e. ˜y(z). Consequen ly, o a ixed alue o γ, la ge alues o c educe he size o
he ob ained in e al a he expense o inc easing he ac ion o ou pu s ha a e
no con ained in he in e al es ima ions. The e o e, gi en γ, he co esponding
alue o cshould be chosen as he la ges alue o c ha gua an ees in he
alida ion se ha he ob ained in e als con ain he ou pu s wi h he desi ed
p obabili y.
F om he discussion abo e, i is clea ha he pa ame e cco esponding o a
pa icula choice o γ > 0 (deno ed cγ) is de e mined by τ. As i is de ailed in
Algo i hm 3, cγis chosen as he la ges alue o c(up o a gi en accu acy ϵ > 0)
ha gua an ees in he alida ion se ha he ob ained con idence in e als con ain
he ou pu s wi h he desi ed p obabili y, ha is, no smalle han 1 −2τ.
Pa ame e γ > 0 can be ob ained by maximizing he likelihood a io which, o a
3.1. Uni a ia e case 37
Algo i hm 3: Op imal alue o c≥0, o gi en γ≥0 and τ∈(0,1)
Da a: τ≥0, γ≥0, cmax >0 and ϵ > 0, Z,Y,ˇ
Yand he alida ion se V.
Resul : cγ.
1cmin = 0;
2while cmax −cmin ≥ϵdo
3c=1
2(cmax +cmin);
4Compu e, using Algo i hm 2, he NVin e als
Ii= [y−
τ(¯zi, γ, c), y+
τ(¯zi, γ, c)], i = 1, . . . , NV.
5Make n+
iol equal o he numbe o iola ions o he uppe cons ain s
¯yi≤y+
τ(¯zi, γ, c), i = 1, . . . , NV,
and n−
iol equal o he numbe o iola ions o he lowe cons ain s
¯yi≥y−
τ(¯zi, γ, c), i = 1, . . . , NV.
6i max{n+
iol, n−
iol}
NV
< τ hen
7cmin =c;
8else
9cmax =c;
10 end i
11 end while
12 cγ=cmin;
38 Chap e 3. P obabilis ic p edic ion egions
speci ic γand co esponding cγ, is de ined as
Lγ=
NV
X
i=1
log ecpγ,c (¯zi,¯yi).
Thus, using ˇ
Y={ˇy1,...,ˇyM}, he op imal alue o γis gi en by
γ∗
τ≈a g max
γ∈Γ
NV
X
i=1
log
e−cγJγ(¯zi,¯yi)
M
P
j=1
e−cγJγ(¯zi,ˇyj)
,(3.4)
whe e Γ is a se con aining all he possible alues conside ed o γ.
Rema k 3.2. O he c i e ia can be used o compu e γ∗
τ. Fo example, γ∗
τcould
be ob ained by minimizing a cos unc ion penalizing he a e age leng h o he in-
e als and/o he a e age p edic ion e o wi h espec o he condi ioned median
in oduced in Rema k 3.1. Howe e , explici ly minimizing he size o he in e als
may ansla e in o an inc eased iola ion a e when he alida ion se has no a
su icien ly la ge numbe o samples, hence no being ep esen a i e enough o he
eal dis ibu ion o y.
3.1.2 Cla i ying example: uni o m dis ibu ion
A sample o 600 poin s in Ris ob ained om a uni o m p obabili y unc ion wi h
suppo [0,1]. One hal o he a ailable poin s is used as a aining se and he
o he hal is used as a es se .
Figu e 3.1 shows he empi ical p obabili y densi y unc ions es ima ed using di -
e en alues o he pa ame e s cand γ. In his case, c= 1.5 and γ= 5 is he pai
ha achie es he bes i o he dis ibu ion p oposed in his example.
3.1.3 Nume ical example: Lo enz a ac o
The Lo enz a ac o is a sys em o ODEs known o ha ing chao ic solu ions
wi h ce ain alues o he pa ame e s o he sys em. The equa ions ha de ine
he sys em a e he ollowing
do
d =σ(p−o)
dp
d =o(ρ−q)−p(3.5)
dq
d =op −βq ,
whe e σ,ρand βa e eal scala pa ame e s. In his example, hese pa ame e s
ake he alues σ= 10, ρ= 28 and β= 8/3. Fu he mo e, in o de o ob ain he
3.1. Uni a ia e case 39
-0.5 0 0.5 1 1.5
c=150, =0
0
0.5
1
1.5
-0.5 0 0.5 1 1.5
c=15, =0.4
0
0.5
1
1.5
-0.5 0 0.5 1 1.5
c=7.5, =1
0
0.5
1
1.5
-0.5 0 0.5 1 1.5
c=1.5, =5
0
0.5
1
1.5
Figu e 3.1: Es ima ed p obabili y dis ibu ion unc ions.
necessa y da a, he ODEs ha e been in eg a ed nume ically wi h a ixed ime s ep
o Ts= 0.1sand ini ial condi ions o0= 1, p0= 1 and q0= 1. He e, i is conside ed
he ask o o ecas ing he one-s ep ahead alue o o, i.e., yk+1 =ok+1, using he
wo p e ious alues o o, ha is, he eg esso ec o will be zk= [yk, yk−1]⊤.
To s a wi h, 2500 da a poin s a e conside ed, no malized in he [0,1] ange.
Di e en sizes o he da a se s Zand Ya e conside ed in his example (200,
350 and 500 poin s). The alida ion se Vconsis s o 1000 da a poin s and o he
1000 da a poin s a e used as a es se , deno ed by T. He e, we conside he
dissimila i y unc ion o ema k 2.3. The se Γ is aken om [0,3] using a 0.1
sampling s ep. On he o he hand, ˇ
Yis ob ained om a g id o equally dis an
poin s in he in e al [−0.1893,1.2298] sampled wi h a 1.4191 ×10−4s ep.
Two di e en echniques will be conside ed as benchma ks. The i s one is quan-
ile eg ession [65], [66], a classical me hod o he es ima ion o condi ioned quan-
iles. The second one is he se -membe ship me hod desc ibed in [49, 50]. This
echnique is a well-known me hod o gene a e in e al bounds o a ime se ies
(usually p oduced om a dynamical sys em). Fo he sake o compa ison, o
gua an ee ha hese bounds con ain he ou pu wi hin a p esc ibed p obabili y,
he pa ame e s ϵ, γ o [49] a e chosen so ha he esul ing empi ical p obabili y
o con aining a sample wi hin he alida ion se Vis no smalle han 1 −2τ.
The nume ical esul s o he p oposed app oach and he wo benchma k echniques
a e shown in able 3.1 o a [o5%, o95%] in e al, ha is, τ= 0.05, and in able
3.2 o [o10%, o90%] (τ= 0.1). The ou pu o he es da a should be con ained
in he i s in e al wi h a p obabili y o 0.9 (0.8 o he second in e al). The
op imal alue o γhas been chosen by maximizing he likelihood unc ion Lγ(see
equa ion (3.4) and igu e 3.3).
40 Chap e 3. P obabilis ic p edic ion egions
Table 3.1: Resul s o he Lo enz A ac o , in e al [o5%, o95%].
P oposed QR SM
NE. P. I. W. E. P. I. W. E. P. I. W.
200 0.9140 2.0578 0.8290 3.0965 0.8960 2.9378
350 0.8990 1.9352 0.8260 3.0550 0.9100 2.4773
500 0.9070 2.0223 0.8410 3.2450 0.9120 2.5671
Table 3.2: Resul s o he Lo enz A ac o , in e al [o10%, o90%].
P oposed QR SM
NE. P. I. W. E. P. I. W. E. P. I. W.
200 0.8060 1.6053 0.7450 2.2776 0.8160 2.4248
350 0.8060 1.6164 0.7270 2.0607 0.7900 1.9797
500 0.8100 1.6195 0.7630 2.4371 0.8100 2.0021
The empi ical p obabili y in he case o he quan ile eg ession clea ly does no
mee he p obabilis ic speci ica ions. In he case o he p oposed app oach and
he se -membe ship me hod, he obse ed ac ion o ou pu s ha all in o he
p edic ed in e als is much close o he desi ed one. No e ha , o all echniques,
he ob ained empi ical p obabili y can be below he desi ed p obabili y. This could
be sol ed elying on a p obabilis ic scaling scheme [63] o p obabilis ic alida ion
schemes [133], [117].
Rega ding he in e al wid h, he p oposed app oach clea ly manages o ob ain
he smalles in e als o each da a se . Fo he [o5%, o95%] case, in compa ison
o he se -membe ship me hod, he mean in e al wid h is 24.35% smalle . Also,
i is 35.96% smalle wi h espec o he mean in e al wid h ob ained by means
o quan ile eg ession. On he o he hand, o he [o10%, o90%] case, he in e als
ob ained wi h he p oposed echnique a e 23.75% smalle han hose ob ained by
means o he se -membe ship me hod and 28.21% smalle han hose ob ained
using quan ile eg ession echniques. Taking in o accoun he empi ical p obabil-
i y alues and he in e al wid hs, i is possible o conclude ha he p oposed
app oach ob ains he bes esul s.
Finally, in igu e 3.2, a ac ion o he es se Tis shown along wi h he compu ed
in e als [o5%, o95%] o he p oposed app oach. No e ha he in e als a e wide
when he e a e end changes in he ou pu . Fu he mo e, igu e 3.3 shows an
example o he alue o he maximum likelihood a io Lγas a unc ion o γ(in
his case o he da a se o 200 poin s and in e al [o5%, o95%]).
3.1.4 Nume ical example: Dow Jones indus ial a e age index
The example p esen ed in he p e ious chap e will be used again o show he
e ec i eness o he p oposed app oach o in e al o ecas ing. The pa ame e s o
3.1. Uni a ia e case 41
Figu e 3.2: Tes se and compu ed in e als o Lo enz A ac o (in e al [o5%, o95%]).
0 0.5 1 1.5 2 2.5 3
-6350
-6300
-6250
-6200
-6150
-6100
L
Figu e 3.3: Maximum likelihood a io as a unc ion o γ.
he algo i hms a e NΩ= 250, M= 1000, ˇy1= 6684.3, ˇyM= 19445, cmax = 15. As
in he p e ious example, he dissimila i y unc ion used is he one co esponding
o ema k 2.3. The se Γ is aken om [0,5] using a 0.5 sampling s ep. The
eg esso is compounded as in chap e 2. The 10- h and 90- h pe cen iles (i.e.
τ= 0.1) co esponds o he desi ed p obabili y o he p ice in e als. No e ha ,
in o de o compu e he in e als wi h he p oposed me hodology, i is needed
o conside each l-s ep p edic ion independen ly, ob aining ldi e en p edic o s,
each one o a ce ain l-s ep o ecas ing.
The esul s a e shown in igu es 3.4 and 3.5. The e, he p ice in e als a e p e-
sen ed as en elopes. Al hough hese in e als a e igh o l= 1, hey become
la ge as linc eases. This is s aigh o wa d due o he ac ha he unce ain y
inc eases as he p edic ion ho izon does. Also, no e ha some imes he eal p ice
is no included wi hin he compu ed p ice in e al. This is cong uen o he ac
ha he p obabili y o belonging o he in e al is 80%.
42 Chap e 3. P obabilis ic p edic ion egions
Figu e 3.4: P ice in e als o 1 o 3 days (5-day EMA).
3.1. Uni a ia e case 43
Figu e 3.5: P ice in e als o 4 and 5 days (5-day EMA).
44 Chap e 3. P obabilis ic p edic ion egions
As a baseline app oach, a quan ile eg ession app oach is chosen o alida e he
ob ained esul s. In able 3.3 i is shown he empi ical p obabili ies and in e al
wid h o he p oposed app oach and he quan ile eg ession scheme. Al hough
he p oposed app oach con ains he eal p ice wi h a highe p obabili y han he
desi ed one, he quan ile eg ession ails o ob ain in e als mee ing he speci ied
p obabili y o any l.
Table 3.3: Empi ical p obabili y and a e age in e al wid h.
Empi ical p obabili y A e age in e al wid h
kP oposed Q. eg ession P oposed Q. eg ession
1 0.8679 0.6006 162.4802 99.8091
2 0.8459 0.6101 315.7218 190.5443
3 0.8553 0.6321 477.8004 287.5774
4 0.8648 0.6761 638.8747 397.3780
5 0.8805 0.6950 813.5661 489.7816
3.2 Mul i a ia e case
In his sec ion, he p oblem o mul idimensional sys ems o ime-se ies is ackled,
p o iding p edic ion egions o he mul idimensional ou pu . As he scheme
p oposed in he p e ious sec ion o uni a ia e ou pu s is based on a nume ical
in eg a ion in a unidimensional space, he gene aliza ion o he mul idimensional
se ing is no i ial because o he well-known complexi y o high-dimensional
nume ical in eg a ion. In wha ollows, i is shown how o ob ain p edic ion
egions in mul idimensional spaces wi hou eso ing o nume ical in eg a ion,
p o iding a undamen al ad an age wi h espec o he p e ious esul s.
Fi s , implici egions compu ed by means o he dissimila i y unc ion a e p o-
posed. A e ha , an ellipsoidal app oxima ion is p oposed. Bo h app oaches a e
es ed in a nume ical example.
3.2.1 Implici egions
In o de o cha ac e ize he desi ed p edic ion egion, wo issues ha e o be ackled:
he choice o he egion cen e and he compu a ion o he egion i sel .
Choosing he cen e
In he p e ious chap e , ˜y(z) was deno ed as he alue o y ha minimizes he
dissimila i y unc ion gi en z(see (2.7)). As a consequence o his, ˜y(z) can be
conside ed as a good op ion o de ine he cen e o he egion ha will be ob ained
as i is he mos likely alue o he ou pu gi en z,Zand Y.
This op imal alue can be ob ained as ˜y(z) = Y λ∗whe e λ∗co esponds o he
3.2. Mul i a ia e case 45
op imal solu ion o he s ic ly con ex p oblem (2.8), ha is
Jγ(z, Z) = min
λ(1),...,λ(N)
(1 −γ)
N
X
i=1
λ2
(i)+γ
N
X
i=1 |λ(i)|
s. . z=
N
X
i=1
λ(i)¯zi
1 =
N
X
i=1
λ(i).
Compu ing he egions
In he p e ious sec ion, i was shown ha he dissimila i y unc ion can be seen
as a so o su oga e o he p obabili y dis ibu ion o ygi en z. Thus, his
p obabili y dis ibu ion peaks a ˜y(z) and dec eases as he dissimila i y unc ion
inc eases. Fo ha eason, he p oposed p edic ion egions a e de ined as hose
poin s o which he dissimila i y unc ion does no exceed mo e han a gi en
ac o αwi h espec o he alue co esponding o he cen al p edic ion ˜y(z),
ha is, J∗
γ(z). This is o mally s a ed in he ollowing de ini ion.
De ini ion 3.3 (P edic ion egion).Fo a gi en z,γ, da a se s Z,Yand a unable
pa ame e α > 1, he p oposed p edic ion egion is de ined as he se
∆(z) = y:Jγ(z, y)≤αJ∗
γ(z),(3.7)
ha is, he poin s y ha ob ain a dissimila i y less o equal o αJ∗
γ(z).
Since he dissimila i y unc ion is con ex in y, he ob ained implici egions a e
con ex and can be used in di e en se ings, e.g. in chance-cons ained op i-
miza ion p oblems. Fo example, in many cases, i is no necessa y o compu e
a p edic ion egion, bu o e i y i a ce ain poin ¯ybelongs o i . This is an
a o dable ask as i su ices o compu e he dissimila i y Jγ(z, ¯y) and check ha
(3.7) holds.
Tuning he hype pa ame e s αand γ
The alue o γ≥0 could be ob ained in such a way ha he p edic ion e o
co esponding o he cen al p edic ion in a alida ion se is minimized. Fo a
gi en γ, smalle alues o αmake he egions smalle . Ou objec i e is o ob ain
he smalles possible egion ha gua an ees ha a poin ¯yj aken om a alida ion
se Vis con ained in he compu ed egion ∆(zj) wi h a p e-speci ied p obabili y
1−τ. Again, as in he uni a ia e case, i is assumed ha he alida ion se V
ep esen s he ue p obabili y dis ibu ion. Then, αis chosen so ha i ul ills
P obV(y∈∆(z)) ≥1−τ,
52 Chap e 3. P obabilis ic p edic ion egions
Table 3.4: A ea o he egions and empi ical p obabili ies (E.P.) ob ained o he p o-
posed me hodology, he app oxima ion wi h ellipsoidal egions, Gaussian p ocesses (GPs),
quan ile eg ession (QR) and in e se eg ession (IR) .
P oposed App oxima ion GPs QR IR
NA ea E.P. A ea E.P. A ea E.P. A ea E.P. A ea E.P.
τ= 0.1250 0.1695 0.9200 0.1885 0.9170 0.2235 0.9250 0.5839 0.9120 0.4331 0.9031
500 0.1551 0.9260 0.1656 0.9170 0.2092 0.9240 0.2645 0.8960 0.4101 0.9043
τ= 0.2250 0.1119 0.8070 0.1176 0.8070 0.1518 0.8450 0.2341 0.8550 0.2466 0.7446
500 0.1053 0.8150 0.1142 0.8260 0.1403 0.8310 0.1590 0.8080 0.2335 0.7454
ou pe o m he baselines conside ed o bo h alues o τand di e en da a se
sizes. Tha is, he p oposed app oach ob ains egions o conside ably smalle
a ea while s ill ul illing he speci ied p obabili y gi en by τ(i.e. E.P. ≥1−τ).
This means ha he p oposed app oach p o ides egions o smalle unce ain y
in compa ison o he baselines conside ed.
3.3 Conclusions
This chap e p esen ed me hods o ob ain in e al p edic ions and p edic ion e-
gions, which can be conside ed as an ex ension o he p e ious chap e whe e only
he expec ed alue o he ou pu o a ime-se ies o a dynamic sys em was ob-
ained. Fi s , a me hod based on ob aining an empi ical condi ioned p obabili y
densi y unc ion was p oposed. Howe e , his me hod u ned ou o be compu-
a ionally expensi e due o he ac ha i equi ed he nume ical in eg a ion o
his p obabili y densi y unc ion o e e y ime ins an . Two nume ical examples
we e p oposed: he o ecas ing o he closing p ice o he Dow Jones, which was
p esen ed in he p e ious chap e , and he p edic ion o he Lo enz a ac o ,
which is a dynamic sys em known o i s chao ic beha iou . Some compa isons
a e made wi h espec o some baseline echniques such as quan ile eg ession and
se -membe ship me hods. In bo h examples, he p oposed app oach ob ained
be e esul s.
In he second hal o he chap e , i was p oposed a me hod ha no only allows
us o ob ain p edic ion egions o mul i a ia e sys ems o ime-se ies bu also
educes conside able he compu a ional bu den. This is due o he ac ha he
second me hod no longe needs he in eg a ion o he empi ical condi ioned densi y
unc ion. Also, a me hod o ob ain ellipsoidal app oxima ions o such egions was
p oposed. Finally, a p edic ion example o a ki e sys em was p oposed. O he
app oaches including GPs, IR, e c. we e conside ed as baselines. I was shown
ha he p oposed app oach and he app oxima ed ellipsoidal egions achie ed he
smalles size while ul illing he desi ed p obabilis ic speci ica ions.
Pa II
K iging-based iden i ica ion
55
Chap e 4
S a e-space k iging o
au onomous sys ems
In pa I, some me hods o make p edic ions o ime-se ies o dynamical sys ems
based on dissimila i y unc ions we e p oposed. These included l-s eps ahead
o ecas ings, in e al p edic ions, p edic ions egions, e c.
F om now on, he objec i e will be o ob ain a dynamic model o a sys em gi en
some pas da a o i s ou pu s and inpu s. In he case o au onomous sys ems
only pas ou pu s will be conside ed, whe eas o non-au onomous sys ems pas
inpu s will be conside ed as well. The s a e-space o his new model will be
compounded o he ec o o weigh s λappea ing in he dissimila i y unc ion
p esen ed in pa I. Two di e en s a egies a e p oposed, a linea ime a ian
app oach based on he weigh ing o he local da a wi hin he dissimila i y unc ion
and a ke nel-based app oach whe e he dissimila i y unc ion is sligh ly modi ied
o accommoda e he ke nel ick. Finally, some nume ical examples a e p esen ed
o show he e ec i eness o he p oposed app oaches.
4.1 Dynamic k iging
Conside an au onomous disc e e nonlinea sys em
xk+1 =h(xk) (4.1a)
yk=g(xk),(4.1b)
whe e kis he ime ins an , xk∈Rnxis he s a e o he sys em, yk∈Rnyis he
ou pu o he sys em, whe eas h(·) and g(·) a e unknown nonlinea unc ions such
ha h(·) : Rnx→Rnxand g(·) : Rnx→Rny.
The objec i e o his sec ion is o ob ain a model o he ou pu s o (4.1). I
is assumed ha he only a ailable da a a e he measu able ou pu s. The e-
g esso zkis de ined as a ime delay embedding ec o zk∈Rnzcon aining he
56 Chap e 4. S a e-space k iging o au onomous sys ems
nppas ou pu s o he sys em. Tha is, zk= [y⊤
k, y⊤
k−1, ..., y⊤
k−np+1]⊤∈Rnz,
whe e nz=npny. Also, no e ha z+
kis deno ed as he successo o zk, i.e.
z+
k= [y⊤
k+1, y⊤
k, ..., y⊤
k−np+2]⊤∈Rnz.
F om now on, assume ha some his o ical da a o he plan is s o ed in a da abase
in he o m o ma ices:
D=¯z1¯z2. . . ¯zN,
D+=¯z+
1¯z+
2. . . ¯z+
N,
whe e N > nzis he numbe o da a poin s, ¯z e e s o a sample o zand ma ix
D+is he successo o D. The indexes o he columns o Dand D+do no e e
o he sample ime, bu o he posi ion in he ma ix. The e o e, ¯zi+1 is no
necessa ily he successo sample o ¯zi. A sample ime k, an es ima ion o he
successo o zk, deno ed as ˜zk+1, can be ob ained by a linea combina ion o he
columns o ma ix D+using he ec o o op imal weigh s λ∗
k∈RNob ained om
an op imiza ion p oblem simila o hose shown in sec ion 2, i.e.
˜zk+1 =D+λ∗
k.
Conside ing de ini ion 2.1 and assuming ha γ= 0, he op imiza ion p oblem
om which λ∗
kis ob ained can be posed as
λ∗
k= a g min
λk
λ⊤
kH1λk(4.2a)
s. . D
1λk=zk
1,(4.2b)
whe e H1∈RN×Nis a posi i e de ini e weigh ing ma ix and 1a ow ec o
wi h all i s componen s equal o 1. Fo cing he componen s o λk o sum one is
equi alen o including a bias e m in he es ima ion p ocess. The simples choice
is o make H1equal o he iden i y ma ix IN(see ema k 2.2). This op imiza ion
p oblem can be ew i en as
λ∗
k= a g min
λk
λ⊤
kH1λk(4.3a)
s. . Cλk=b , (4.3b)
wi h
C=D
1, b =zk
1.
In o de o gua an ee ha any poin in Rnz+1 can be exp essed as a linea com-
bina ion o he columns o C, i is assumed ha ma ix Cis ull ow ank. This
equali y cons ained quad a ic p oblem has an analy ic solu ion ha can be ob-
ained compu ing he Lag angian and i s de i a i e:
L(λk, ν) = λ⊤
kH1λk+ν⊤(Cλk−b)
4.2. Linea s a e-space k iging 57
d
dλL(λk, ν) = 2H1λk+C⊤ν ,
whe e νis he dual a iable associa ed wi h he equali y cons ain . F om he
Ka ush-Kuhn-Tucke (KKT) condi ions:
2H1λ∗
k+C⊤ν∗= 0 ,(4.4)
which leads o
λ∗
k=−H−1
1C⊤ν∗
2.
P e-mul iplying his equali y by C, and aking in o accoun ha Cλ∗
k=b, he
ollowing is ob ained
ν∗=−2CH−1
1C⊤−1b,
which applied o equa ion (4.4) yields
λ∗
k=H−1
1C⊤CH−1
1C⊤−1zk
1.
In o de o p edic zk+d, wi h d > 1, one could use his app oach in a ecu si e
way. Tha is, he i- h ahead p edic ion ˜zk+icould be used o compu e
λ∗
k+i=H−1
1C⊤CH−1
1C⊤−1˜zk+i
1,
and hus ob aining ˜zk+i+1 =D+λ∗
k+i. In he nex sec ion, i is p oposed a modi-
ica ion o his nai e ecu si e me hod. The no el me hodology elies on a ime-
a ying s a e-space modelling o he op imal weigh ing ec o pa ame e λ∗
k.
4.2 Linea s a e-space k iging
Suppose ha he p edic ion ˜zk+1 o zk+1 is ob ained om ˜zk+1 =D+λ∗
k, whe e
he sum o he componen s o λ∗
kis assumed o be equal o one. In o de o
model how he dynamics o he op imal ec o o weigh s λ∗
k+1 depends on λ∗
k,
a egula iza ion e m is added o op imiza ion p oblem (4.2) ha penalizes he
di e ence be ween λ∗
k+1 and λ∗
k. In his way, ec o λ∗
k+1 no only ul ills he
equi ed equali y cons ain s, bu also does no depa excessi ely om λ∗
k. This
will educe he sensi i i y o noise o he iden i ied dynamics. Thus, gi en ˜zk+1,
λ∗
k+1 is ob ained om
λ∗
k+1 = a g min
λk+1
(λk+1 −λ∗
k)⊤H2(λk+1 −λ∗
k) + λ⊤
k+1H1λk+1
s. . D
1λk+1 =˜zk+1
1,
58 Chap e 4. S a e-space k iging o au onomous sys ems
whe e H2∈RN×Nis chosen as he iden i y ma ix mul iplied by a ce ain scala
ha could be selec ed by c oss- alida ion [1, §16.5]. Because o he assump ions
on λ∗
k, he p e ious op imiza ion p oblem can be ew i en as
λ∗
k+1 = a g min
λk+1
(λk+1 −λ∗
k)⊤H2(λk+1 −λ∗
k) + λ⊤
k+1H1λk+1
s. . D
1λk+1 =D+
1λ∗
k.
Thus, he p oblem becomes
λ∗
k+1 = a g min
λk+1
(λk+1 −λ∗
k)⊤H2(λk+1 −λ∗
k) + λ⊤
k+1H1λk+1 (4.5a)
s. . Cλk+1 =C+λ∗
k,(4.5b)
wi h
C=D
1, C+=D+
1.
No e ha λ∗
k+1 is de e mined only by λ∗
kand ma ices Cand C+. Op imiza ion
p oblem (4.5) can be ew i en as
λ∗
k+1 = a g min
λk+1
1
2λ⊤
k+1Hλk+1 + ⊤λk+1
s. . Cλk+1 =b ,
wi h H= 2(H1+H2), =−2H2λ∗
kand b=C+λ∗
k. Also, no e ha he cons an
e m λ∗
k
⊤H2λ∗
kis emo ed because i does no a ec he solu ion λ∗
k+1. The
Lag angian o his p oblem is gi en by
L(λk+1, ν) = 1
2λ⊤
k+1Hλk+1 + ⊤λk+1 +ν⊤(Cλk+1 −b),
whe e νis he dual a iable associa ed wi h he equali y cons ain . The de i a i e
o he Lag angian is
d
dλk+1 L(λk+1, ν) = Hλk+1 + +C⊤ν .
In he op imum, he de i a i e ul ills he KKT condi ions [139, §10.1.1]. Tha is,
Hλ∗
k+1 + +C⊤ν∗= 0 ,
and hus
λ∗
k+1 =−H−1 −H−1C⊤ν∗.(4.7)
P e-mul iplying bo h sides o las equali y by Cyields
b=Cλ∗
k+1 =−CH−1 −CH−1C⊤ν∗,
4.2. Linea s a e-space k iging 59
and hus ν∗= (CH−1C⊤)−1(−CH−1 −b). Subs i u ing his in o equa ion (4.7),
he ollowing exp ession o λ∗
k+1 is ob ained
λ∗
k+1 =H−1C⊤CH−1C⊤−1CH−1 +b−H−1 .
Taking in o accoun ha =−2H2λ∗
kand b=C+λ∗
k, he s a e-space equa ion o
λ∗
kcan be w i en as
λ∗
k+1 =Aλ∗
k,
wi h
A= 2H−1H2+H−1C⊤(CH−1C⊤)−1(C+−2CH−1H2).
No e ha his means ha λk ollows linea dynamics. Thus, a new model o he
ou pu s o sys em (4.1) has been ob ained using his o ical da a o hese ou pu s.
This new au onomous sys em allows us o compu e he nex alues o λand z.
No e ha , as only he i s e m o zkis needed (i.e. he e m co esponding o
yk), he model can be posed as
λ∗
k+1 =Aλ∗
k
yk=Y λ∗
k,
whe e Ydeno es a ma ix compounded o only he i s ows o D, ha is, Yis
a ma ix con aining he samples ¯yi.
4.2.1 Ini ial condi ion
The ini ial ec o o op imal weigh s λ∗
0is compu ed by means o he op imiza ion
p oblem in (4.2) subs i u ing kby 0, ha is
λ∗
0= a g min
λ0
λ⊤
0H1λ0
s. . D
1λ0=z0
1.
4.2.2 Local-da a app oach
No e ha he p e ious model is linea and ime-in a ian as he ma ix Ais
cons an . Howe e , i is possible o weigh he poin s in he da a se wi h espec
o z, which would encou age he use o local da a and hus p o ide be e esul s
when iden i ying nonlinea sys ems.
This can be done by choosing H1app op ia ely. In ema k 2.2, i was shown ha
i is possible o conside a ec o ωwi hin he dissimila i y unc ion o weigh
di e en elemen s in he da a se . He e, he squa ed Euclidean dis ance is chosen
o measu e he dissimila i y o a ce ain z o each poin o he da a se . Tha is,
60 Chap e 4. S a e-space k iging o au onomous sys ems
ω(z) =
(z−¯z1)⊤(z−¯z1)
.
.
.
(z−¯zN)⊤(z−¯zN)
,(4.8)
and hus
H1= diag(ω(z)) ,
whe e diag(·) deno es a diagonal ma ix RN×RNwhose non-ze o en ies a e he
componen s o he inpu ec o . No e ha , because o his change, his ec o
ω(z) needs o be compu ed a each ime ins an k, leading o a di e en ma ix
A o e e y k, and hus leading o LTV dynamics o λ, ha is
λ∗
k+1 =Akλ∗
k
yk=Y λ∗
k.
4.3 Ke nel-based s a e-space k iging
Ke nels unc ions a e widely used in he machine lea ning ield. Fo example, Sup-
po Vec o Machines (SVM) a e supe ised lea ning models ha classi y linea ly
sepa able da a. Tha is, gi en a cloud o da a poin s (each poin belonging o
a ce ain class), he p oblem o classi ica ion is de ined as inding an hype plane
ha di ide he da a in o wo se s. Howe e , when conside ing nonlinea ela ions
wi hin he da a, i is no possible o ob ain an hype plane ha sepa a es he da a.
In o de o be able o classi y his kind o da a, i is needed o p ojec his da a
in o a high-dimensional space whe e i may become linea ly sepa able.
Thanks o he so-called ke nel ick (see chap e 3 in [150]), i is possible o ope a e
in a high-dimensional ea u e space wi hou compu ing he coo dina es o such
space. Ins ead, i is only needed o compu e he inne p oduc o he images o
all pai s o he da a samples, which bypasses he compu a ion o he coo dina es
o he ea u e space ha may be cumbe some o e en impossible.
Thus, ins ead o using local da a as i was p esen ed in he p e ious sec ion, one
could eso o he use o ke nels in o de o model he nonlinea dynamics o
he sys em. Also, he use o ke nels will allow us o compu e he ma ices o he
sys em only once, unlike he local da a app oach whe e he ma ices needed o be
ecompu ed a each sample ime.
In o de o include he ke nels, he op imiza ion p oblem is modi ied, becoming
λ∗
k+1 = a g min
λk+1
(λk+1 −λ∗
k)⊤H2(λk+1 −λ∗
k) + λ⊤
k+1H1λk+1
+
N
X
i=1
φ¯ziλk+1,(i)−
N
X
i=1
φ¯z+
iλ∗
k,(i)
2
Σ−1
φ
s. . 1λk+1 = 1 ,
4.3. Ke nel-based s a e-space k iging 61
whe e φ(·) : Rnz→ H e e s o a nonlinea ope a o ha maps Rnzin o a p ob-
ably high dimensional space H,φ¯ziand φ¯z+
ideno e φ(¯zi) and φ(¯z+
i) espec i ely
and Σφis a posi i e de ini e ma ix o app op ia e dimensions. In wha ollows,
i is shown ha we do no need a p ecise knowledge o φ(·) o compu e λk+1 as
i su ices o compu e, o a gi en pai a∈Rnzand b∈Rnz, he p oduc
⟨φa, φb⟩=φaΣ−1
φφb.
No e ha he p e ious linea ha d cons ain on zkhas been changed o a penal y
e m on a high dimensional space by means o he ke nel ick as i is no longe a
linea cons ain . Assuming ha some da a se s o he e alua ion o φ(·) o e he
ime delay embeddings o Dand D+a e a ailable (which due o he ke nel ick
a e no eally necessa y), i would be possible o deno e hem as
φ¯z=φ¯z1φ¯z2. . . φ¯zN, φ¯z+=hφ¯z+
1φ¯z+
2. . . φ¯z+
Ni.
Thus, he p e ious p oblem could be w i en in ma ix o m as
λ∗
k+1 = a g min
λk+1
(λk+1 −λ∗
k)⊤H2(λk+1 −λ∗
k) + λ⊤
k+1H1λk+1
+∥φ¯zλk+1 −φ¯z+λ∗
k∥2
Σ−1
φ
s. . 1λk+1 = 1 .
Now, ope a ing wi h he e m ∥φ¯zλk+1 −φ¯z+λ∗
k∥2
Σ−1
φ, he ollowing exp ession is
ob ained
∥φ¯zλk+1 −φ¯z+λ∗
k∥2
Σ−1
φ=λ⊤
k+1 φ⊤
¯zΣ−1
φφ¯zλk+1 −2λ⊤
k+1 φ⊤
¯zΣ−1
φφ¯z+λ∗
k
+λ∗
k
⊤φ⊤
¯z+Σ−1
φφ¯z+λ∗
k.
Again, he cons an e m λ∗
k
⊤φ⊤
¯z+Σ−1
φφ¯z+λ∗
kis disca ded because i does no
a ec he alues o λ∗
k+1, leading o he ollowing op imiza ion p oblem
λ∗
k+1 = a g min
λk+1
1
2λ⊤
k+1Hλk+1 + ⊤λk+1 (4.12a)
s. . 1λk+1 = 1 .(4.12b)
wi h H= 2(H1+H2)+2φ⊤
¯zΣ−1
φφ¯z, =−2(H2+φ⊤
¯zΣ−1
φφ¯z+)λ∗
k.
No e ha he ke nel ela ed e ms can be compu ed because only c oss p oduc s
appea in he a o emen ioned equa ions. These e ms would be compu ed as
φ⊤
¯zΣ−1
φφ¯z=
⟨φ¯z1, φ¯z1⟩ ⟨φ¯z1, φ¯z2⟩. . . ⟨φ¯z1, φ¯zN⟩
⟨φ¯z2, φ¯z1⟩ ⟨φ¯z2, φ¯z2⟩. . . ⟨φ¯z2, φ¯zN⟩
.
.
..
.
..
.
.
⟨φ¯zN, φ¯z1⟩ ⟨φ¯zN, φ¯z2⟩. . . ⟨φ¯zN, φ¯zN⟩
,
68 Chap e 4. S a e-space k iging o au onomous sys ems
0 5 10 15
-5
0
5
o
K-SSK
Real
Measu ed Ou pu
0 5 10 15
-5
0
5
p
0 5 10 15
Time (s)
-5
0
5
l
Figu e 4.2: Fo ecas ing he noisy R¨ossle a ac o wi h kalman il e ing.
LD-SSK K-SSK GP NARX K-DMD RC
62.909 37.5912 2.591 14.872 0.036 0.868
Table 4.3: A e age online compu a ional ime in milliseconds (R¨ossle a ac o ).
4.6 Conclusions
This chap e p esen ed he s a e-space k iging me hod o au onomous sys ems.
The p oposed echnique allowed us o ob ain a model o he sys em by means o
pas da a o he p ocess. This could be done by manipula ing app op ia ely he
dissimila i y unc ion p esen ed in pa I and by ob aining he explici solu ion
o he esul ing op imiza ion p oblems. Also, i was shown ha a kalman il e
can be used o imp o e he pe o mance o he p oposed models. Finally, wo
nume ical examples we e p oposed in o de o show he e ec i eness o he SSK
app oaches. Bo h he LD-SSK and he K-SSK a ained be e pe o mance han
o he exis ing machine lea ning models.
In he nex chap e , he p oposed me hodology will be ex ended o ackle non-
au onomous sys em, which will allow us o de elop MPC con olle s as well.
69
Chap e 5
S a e-space k iging o
non-au onomous sys ems
The p e ious chap e in oduced he concep o s a e-space k iging o au onomo-
us sys ems and, by means o nume ical esul s, i was shown ha he pe o mance
is, i no be e , compa able o many o he o ecas ing me hods. Howe e , in o de
o be able o use he p oposed app oaches in a con ol scheme, i is needed o
in oduce he inpu e m in he SSK o mula ion.
In his chap e , bo h me hodologies p esen ed be o e will be ex ended o ackle
non-au onomous sys ems. I is also shown ha he ke nel-based SSK is mo e
sui able o con ol due o he ac ha he sys em ma ices a e compu ed only
once, unlike he local da a app oach.
5.1 Non-au onomous linea SSK
Conside a non-au onomous disc e e nonlinea sys em
xk+1 =h(xk, uk) (5.1a)
yk=g(xk),(5.1b)
whe e kis he ime ins an , xk∈Rnxis he s a e o he sys em, uk∈Rnuis he
inpu o he sys em, yk∈Rnyis he ou pu o he sys em and h(·) and g(·) a e
unknown nonlinea unc ions such ha h(·) : Rnx×nu→Rnxand g(·) : Rnx→Rny.
Unlike in chap e 4, i is assumed ha he ec o o weigh s λ∗
k ul ill he ollowing
cons ain s:
zk=D+λ∗
k,
1 = 1λ∗
k.
70 Chap e 5. S a e-space k iging o non-au onomous sys ems
Fu he mo e, i is imposed ha λ∗
k+1 mus be able o compu e zkand ukas an
a ine combina ion o Dand U. Thus, i mus sa is y he cons ain s
D
1
U
λ∗
k+1 =
zk
1
uk
.
whe e U=¯u1¯u2. . . ¯uN∈ Rnu×Nis he da a se o con ol ac ions, i.e. ¯ui
a e pas samples o he inpu u. This se o cons ain s is simila o he cons ain s
p esen ed in he p e ious chap e bu aimed o ob ain he successo o λ∗
k+1, as
a unc ion o λ∗
kand uk, ha is, o conside non-au onomous sys ems. Thus, he
p oposed op imiza ion p oblem leading o he s a e equa ion is
λ∗
k+1 = a g min
λk+1
(λk+1 −λ∗
k)⊤H2(λk+1 −λ∗
k) + λ⊤
k+1H1λk+1 (5.3a)
s. . Cλk+1 =C+λ∗
k+0(nz+1)×nu
Inuuk,(5.3b)
whe e 0(nz+1)×nuis a ma ix o ze os compounded o nz+ 1 ows and nucolumns
C=D⊤1⊤U⊤⊤and C+=hD+⊤1⊤0⊤i⊤.No e ha ukis he
cu en alue o he sys em inpu , no o be con used wi h ¯u, which a e pas
alues o uks o ed in he da a base U.
This p oblem can be w i en in canonical o m as
λ∗
k+1 = a g min
λk+1
1
2λ⊤
k+1Hλk+1 + ⊤λk+1
s. . Cλk+1 =b ,
wi h H= 2(H1+H2), =−2H2λ∗
kand b=C+λ∗
k+0nu×(nz+1) Inu⊤uk.
No e ha he cons an e m is disca ded because i does no a ec he alue o
λ∗
k+1. By means o he KKT condi ions, his p oblem has he solu ion
λ∗
k+1 =−H−1 +H−1C⊤CH−1C⊤−1CH−1 +b.
Subs i u ing and b, he ollowing s a e-space equa ion is ob ained
λ∗
k+1 =Aλ∗
k+Buk,
yk=Y+λ∗
k.
whe e Y+deno es a ma ix compounded o only he i s ny ows o D+and
A= (2H−1H2+H−1C⊤(CH−1C⊤)−1C+−2CH−1H2),
B=H−1C⊤(CH−1C)−10nu×(nz+1) Inu⊤.
As he exp ession o he dynamics o λhas been ob ained, he ou pu ycan be
easily ob ained o any ime ins an kassuming ha he inpu s a e gi en.
5.2. Non-au onomous ke nel-based SSK 71
5.1.1 Ini ial condi ion
I s ill emains o show how o ob ain he ini ial alue λ∗
0. Fo ha pu pose,
conside he ollowing op imiza ion p oblem
λ∗
0= a g min
λ0
λ⊤
0H1λ0
s. . D+
1λ0=z0
1.
whe e z0is he ini ial alue o he ime delay embedding.
5.2 Non-au onomous ke nel-based SSK
Taking in o accoun ke nel unc ions, he op imiza ion p oblem in (5.3) becomes
he ollowing
λ∗
k+1 = a g min
λk+1
(λk+1 −λ∗
k)⊤H2(λk+1 −λ∗
k) + λ⊤
k+1H1λk+1
+∥φ¯zλk+1 −φ¯z+λ∗
k∥2
Σ−1
φ(5.6a)
s. . U
1λk+1 =uk
1.(5.6b)
As in he p e ious chap e , he ha d cons ain becomes a penal y e m in a high
dimensional space due o he ke nel ick [150]. Now, ope a ing wi h he e m
∥φ¯zλk+1 −φ¯z+λ∗
k∥2
Σ−1
φ, he ollowing exp ession is ob ained
∥φ¯zλk+1 −φ¯z+λ∗
k∥2
Σ−1
φ=λ⊤
k+1φ⊤
¯zΣ−1
φφ¯zλk+1 −2λ⊤
k+1φ⊤
¯zΣ−1
φφ¯z+λ∗
k
+λ∗
k
⊤φ⊤
¯z+Σ−1
φφ¯z+λ∗
k,
Disca ding he cons an e m λ∗
k
⊤φ⊤
¯z+Σ−1
φφ¯z+λ∗
k, we ob ain he ollowing op i-
miza ion p oblem
λ∗
k+1 = a g min
λk+1
1
2λ⊤
k+1Hλk+1 + ⊤λk+1
s. . Tλk+1 =0⊤
1+Inu
0uk.
wi h H= 2(H1+H2)+2φ⊤
¯zΣ−1
φφ¯z, =−2(H2+φ⊤
¯zΣ−1
φφ¯z+)λ∗
k,T=U
1.
No e ha he ke nel ela ed e ms can be compu ed because only c oss p oduc s
appea in he a o emen ioned equa ions. These e ms would be compu ed as
φ⊤
¯zΣ−1
φφ¯z=
⟨φ¯z1, φ¯z1⟩ ⟨φ¯z1, φ¯z2⟩. . . ⟨φ¯z1, φ¯zN⟩
⟨φ¯z2, φ¯z1⟩ ⟨φ¯z2, φ¯z2⟩. . . ⟨φ¯z2, φ¯zN⟩
.
.
..
.
..
.
.
⟨φ¯zN, φ¯z1⟩ ⟨φ¯zN, φ¯z2⟩. . . ⟨φ¯zN, φ¯zN⟩
,
72 Chap e 5. S a e-space k iging o non-au onomous sys ems
φ⊤
¯zΣ−1
φφ¯z+=
⟨φ¯z1, φ¯z+
1⟩ ⟨φ¯z1, φ¯z+
2⟩. . . ⟨φ¯z1, φ¯z+
N⟩
⟨φ¯z2, φ¯z+
1⟩ ⟨φ¯z2, φ¯z+
2⟩. . . ⟨φ¯z2, φ¯z+
N⟩
.
.
..
.
..
.
.
⟨φ¯zN, φ¯z+
1⟩ ⟨φ¯zN, φ¯z+
2⟩. . . ⟨φ¯zN, φ¯z+
N⟩
.
Applying he KKT condi ions, he solu ion o he op imiza ion p oblem (5.6) is
ob ained;
λ∗
k+1 =Aλ∗
k+Buk+c(5.8a)
yk=Y+λ∗
k,(5.8b)
wi h
A= 2H−1(IN−T⊤(TH−1T⊤)−1TH−1)(H2+φ⊤
¯z Σ−1
φφ¯z+),
B=H−1T⊤(TH−1T⊤)−1Inu
0,
c=H−1T⊤(TH−1T⊤)−10⊤
1.
Rema k 5.1. No e ha adding he inpu o he op imiza ion p oblem is compa ible
wi h bo h he Local-da a app oach and he ke nel-based app oach, leading o a
amily o di e en p edic o s. Howe e , in he local-da a app oach, i is needed
o compu e he ma ices Aand Ba each ime ins an k, becoming cumbe some
when conside ing MPC con ol p oblems. Fo ha eason, in he ollowing, i is
only conside ed he ke nel-based SSK.
Rema k 5.2. No e ha he kalman il e o he non-au onomous SSK is equi -
alen o he one p oposed in he p e ious chap e . In his case, equa ion (4.14)
becomes
˜
λ∗
k+1 =Aˆ
λ∗
k+Buk+c . (5.9)
5.2.1 Ini ial condi ion
As in he non-au onomous linea SSK, an ini ial alue λ∗
0is needed. This ini ial
condi ion can be ob ained om he ollowing op imiza ion p oblem
λ∗
0= a g min
λ0
λ⊤
0H1λ0+∥φz0−φ¯z+λ0∥2
Σ−1
φ
s. . 1⊤λ= 1 .
Ope a ing wi h ∥φz0−φ¯z+λ0∥2
Σ−1
φ, we ob ain
λ∗
0= a g min
λ0
1
2λ⊤
0Hλ0+ ⊤λ0
s. . 1⊤λ0= 1 ,
5.3. Applica ion o MPC 73
wi h
H= 2H1+ 2φ⊤
¯z+Σ−1
φφ¯z+, =−2φ⊤
¯z+Σ−1
φφz0,
φ⊤
¯z+Σ−1
φφ¯z+=
⟨φ¯z+
1, φ¯z+
1⟩ ⟨φ¯z+
1, φ¯z+
2⟩. . . ⟨φ¯z+
1, φ¯z+
N⟩
⟨φ¯z+
2, φ¯z+
1⟩ ⟨φ¯z+
2, φ¯z+
2⟩. . . ⟨φ¯z+
2, φ¯z+
N⟩
.
.
..
.
..
.
.
⟨φ¯z+
N, φ¯z+
1⟩ ⟨φ¯z+
N, φ¯z+
2⟩. . . ⟨φ¯z+
N, φ¯z+
N⟩
,
φ⊤
¯z+Σ−1
φφz0=
⟨φ¯z+
1, φz0⟩
⟨φ¯z+
2, φz0⟩
.
.
.
⟨φ¯z+
N, φz0⟩
.
5.3 Applica ion o MPC
Assuming ha we ha e an a bi a y nonlinea sys em in a disc e e ime se ing
like he sys em in equa ion (5.1), hen, he objec i e o an MPC con olle is o
s ee he s a e x o an equilib ium poin (xs, us) ha i is called he e e ence o
he con olle . The con ol ac ions applied o he sys em o achie e his objec i e
uka e compu ed by means o an op imiza ion p oblem.
The ob ained op imal inpu s depend on bo h he ac ual s a e o he plan and
he op imiza ion c i e ia. This s ep cos unc ion is designed o penalize he
de ia ion o he s a e and he inpu s o he e e ence a each ime ins an k o
a ce ain p edic ion ho izon Np. In o de o ensu e s abili y o he con olle , a
e minal cos unc ion is usually added o he p e iously de ined s ep cos unc ion.
Deno ing V(·) as he o al cos unc ion, uas he sequence o inpu s {ui}k+Np−1
i=k,
xas he sequence o p edic ed s a es {xi|k}Np
i=0,xi|kas he p edic ion o xk+imade
a ins an kand conside ing ha he sys em may ha e cons ain s in he inpu s
and in he s a es so ha x∈ X and u∈ U, he op imal con ol inpu s u∗a e
ob ained by sol ing he ollowing op imiza ion p oblem
u∗= a g min
x,uV(x,u) (5.11a)
s. . sys em model (5.11b)
xi|k∈ X ∀i= 0, . . . , Np(5.11c)
uk+i∈ U ∀i= 0, . . . , Np−1.(5.11d)
F om his sequence o op imal inpu s u∗, only he i s componen will be applied
o he sys em, disca ding he es as i is usual in MPC due o he eceding ho izon
s a egy.
In his sec ion, a acking MPC con olle [124, 125] ha uses he K-SSK model is
p esen ed in his sec ion. The main di e ence o he acking MPC wi h espec o
74 Chap e 5. S a e-space k iging o non-au onomous sys ems
adi ional MPC is ha he e e ence o be acked is conside ed as an addi ional
decision a iable in he op imiza ion p oblem, ha is, i becomes an a i icial
e e ence. In o de o en o ce ha he a i icial e e ence con e ges o he a ge
e e ence, an addi ional cos is added o he MPC cos unc ion, penalising he
de ia ion be ween hem. Among he many ad an ages o his o mula ion, he ac
ha he ecu si e easibili y is gua an eed o any change o he desi ed e e ence
and a signi ican ly la ge domain o a ac ion o sho p edic ion ho izons a e
p obably he mos impo an ones.
The cos unc ion in he p oposed con olle is compounded o h ee ing edien s:
•A s ep cos unc ion ls(·,·) o penalize acking e o . T acking e o wi h
espec o a ce ain e e ence is ackled by means o a change o a iables
˘y= ˜y−ys, ˘u=u−uswhe e ˜yis a p edic ion o y,ysis he a i icial
ou pu e e ence and usis he a i icial inpu e e ence. He e, a quad a ic
cos is conside ed. This cos penalizes he dis ance o he a i icial inpu
and ou pu e e ence by means o some weigh ing ma ices o app op ia e
dimensions Qand R(i.e. Q∈Rny×nyand R∈Rnu×nu), i.e.
ls(˘y, ˘u) = ˘y⊤Q˘y+ ˘u⊤R˘u.
•An o se unc ion penalizing he di e ence be ween he a i icial ou pu ys
and he a ge desi ed e e ence,
lo(ys, )=(ys− )⊤O(ys− ).
Unde some mild condi ions [124], i is p o en ha he a i icial e e ence
con e ges o he ue e e ence as ime goes by.
•A weigh ed e minal cos unc ion. This unc ion measu es he closeness o
he e minal s a e λ∗
Np|k o he a i icial s eady s a e λ∗
s. Weigh ing app o-
p ia ely he e minal cos allows us o omi he e minal equali y cons ain
[125], simpli ying he design o he con olle .
l (λ∗
Np|k, λ∗
s) = γλ∗
Np|k−λ∗
s⊤Pλ∗
Np|k−λ∗
s,
whe e γ≥1.
Deno ing y∈RnyNpand u∈Rnu(Np+1) as
y= [y⊤
0|k, . . . , y⊤
Np−1|k]⊤,[u⊤
k, . . . , u⊤
k+Np−1, u⊤
s]⊤,
i is possible o de ine a o al cos unc ion VNp(y,u, , λ∗
k) as he sum o he
a o emen ioned h ee unc ions o a ini e p edic ion ho izon Np
VNp(y,u, , λ∗
k) =
Np−1
X
i=0
ls(˘yi|k,˘uk+i) + lo(ys(us), ) + l (λ∗
Np|k(λ∗
k,u), λ∗
s(us)).
5.3. Applica ion o MPC 75
Thus, an MPC con ol p oblem wi h inequali y cons ain s in he ou pu s and
box cons ain s in he inpu s is p esen ed he e
u∗= a g min
y,u,λNp|k
VNp(y,u, , λ∗
k) (5.12a)
s. . λ∗
i+1|k=Aλ∗
i|k+Buk+i+c∀i= 0, . . . , Np−1 (5.12b)
yi|k=Y+λ∗
i|k∀i= 0, . . . , Np(5.12c)
λ∗
s=Aλ∗
s+Bus+c(5.12d)
ψ yi|k≤δ∀i= 0, . . . , Np(5.12e)
umin ≤uk+i≤umax,∀i= 0, . . . , Np−1 (5.12 )
which is a pa ame ic quad a ic op imiza ion p oblem whose pa ame e s a e
and λ∗
k. Appendix A shows how o pose his p oblem in canonical o m.
F om his op imiza ion p oblem, a sequence o op imal con ol ac ions uis ob-
ained. Howe e , due o he eceding ho izon scheme cha ac e is ic o any MPC
con olle , only he i s componen is applied o he sys em, compu ing a whole
new sequence a nex ime ins an .
5.3.1 Nominal s abili y analysis
Fo he nominal s abili y analysis, i is assumed ha he e a e no misma ches
be ween he dynamics o he eal sys em and he ob ained p edic ion model.
Fu he mo e, he s a e ec o λ∗
kis assumed o be known. This implies ha he
kalman il e is no necessa y, and hus i is no aken in o accoun . Fi s , no e
ha he p oposed model is no s ic ly linea wi h espec o he weigh s λk.
Howe e , aking in o accoun ha
λ∗
k+1 =Aλ∗
k+Buk+c
λ∗
s=Aλ∗
s+Bus+c,
and sub ac ing he equa ions, i is clea ha
(λ∗
k+1 −λ∗
s) = A(λ∗
k−λ∗
s) + B(uk−us).
Making
˘
λ∗
k=λ∗
k−λ∗
s,˘uk=uk−us,
i is easy o see ha he dynamics o ˘
λ∗
ka e linea , ha is
˘
λ∗
k+1 =A˘
λ∗
k+B˘uk.
Now, some addi ional assump ions a e made in o de o p o e he nominal s abili y
o he con olle :
76 Chap e 5. S a e-space k iging o non-au onomous sys ems
Assump ion 5.3. The sys em desc ibed in (5.8) is obse able and he pai (A, B)
is s abilizable.
Assump ion 5.4. Q,Rand Oa e posi i e de ini e ma ices.
Assump ion 5.5. I exis s a s abilizing ma ix Kso ha he ma ix (A+BK)
is Hu wi z.
Assump ion 5.6. I exis s a ma ix Pso ha he Lyapuno equa ion
(A+BK)⊤P(A+BK)−P=−(Q+K⊤RK) (5.13)
is ul illed.
Assump ion 5.7. I exis s a leas one easible equilib ium poin sa is ying he
cons ain s o he con olle .
The p e ious assump ions makes possible o es ablish he ollowing closed-loop
s abili y lemma.
Lemma 5.8 (Closed-loop s abili y).The sys em (5.8) con olled wi h he p o-
posed MPC con olle (5.12) con e ges asymp o ically o he desi ed e e ence. In
he case whe e he desi ed e e ence is un eachable, he sys em con e ges o he
easible e e ence which minimizes he o se cos while main aining he asymp o ic
s abili y.
P oo . Assuming ha he assump ions a e sa is ied, he p oo ollows om [124]
and [125]. In [124], i is p o en he asymp o ic s abili y o he acking con olle
o linea sys ems using a e minal cos e m and a e minal cons ain . Then, in
[125], i is p o en ha he s abili y can be main ained e en i he e is no e minal
cons ain . In ha case, inc easing he weigh ing o he e minal cos makes he
domain o a ac ion o g ow. ■
5.3.2 Robus s abili y analysis
In his sec ion, i is conside ed ha modeling e o s may appea , due o he
ac ha he model o he sys em is no pe ec . Fo his pu pose, he Robus ly
asymp o ically s abili y (RAS) no ion showcased in [106] is chosen. Deno e eas
measu emen e o s and das addi i e dis u bances. Also, deno e eand das
sequences o Npelemen s o eand d. Then, he de ini ion o RAS is he ollowing.
5.4. Examples 77
De ini ion 5.9. The o igin o he closed loop nonlinea sys em xk+1 =
h(xk, κ(xk)) is conside ed o be RAS in he in e io o a se Fwi h espec o
bo h measu emen e o s and addi i e dis u bances i i is possible o ind a KL
unc ion β[152] and o each ϵ > 0 and compac se C ⊂ F he e exis s δ > 0
such ha
1. max(d)≤δ, max(e)≤δ.
2. Any admissible ajec o y is bounded by β(|x|, k) + ϵ.
Lemma 5.10 (Robus asymp o ically s abili y).The sys em (5.8) joined oge he
wi h he kalman il e obse e and con olled wi h he p oposed MPC con olle
(5.12) is RAS wi h espec o measu emen noise and addi i e dis u bances.
P oo . P oposi ion 8 in [106] es ablishes some su icien condi ions o gua an ee
RAS. Ac ually, i is only needed o p o e he con inui y o he Lyapuno unc-
ion. He e, in he p oposed con olle , aking in o accoun ha he p edic ion
model is linea , hen, he pa ame ic quad a ic op imiza ion p oblem is con ex
and he a o emen ioned condi ion holds. Thus, he sys em is RAS wi h espec
o measu emen noise and addi i e dis u bances. ■
5.4 Examples
This sec ion p esen s wo applica ion examples o he p oposed con olle . Fi s ,
he applica ion o a simula ed single-inpu single-ou pu sys em, a con inuously
s i ed ank eac o , is p esen ed, ollowed by he applica ion o a labo a o y
empe a u e con ol equipmen . This is a mul i a iable sys em wi h wo inpu s
and wo ou pu s. Some conside a ions a e aken in o accoun o bo h examples
in o de o gua an ee ha he assump ions a e ul illed:
1. The models ob ained o bo h sys ems a e open-loop s ables, i.e. all he
eigen alues in Aa e less han he uni y. Thus, hei modes a e s able and
hence he sys em is s abilizable (assump ion 5.3).
2. Ma ices Q,Rand Oa e chosen as he iden i y ma ix o app op ia e di-
mensions imes some posi i e cons an , hus hey a e posi i e de ini e (as-
sump ion 5.4).
3. Kand Pa e compu ed using he Ma lab “dlq ” unc ion. This e u ns a
s abilizing gain Kand he ma ix P ha sa is ies he Lyapuno equa ion
(5.13) (assump ions 5.5 and 5.6).
4. Taking in o accoun he p e ious s a emen s and ha he ollowing examples
only conside box cons ain s in he inpu s, i is easy o see ha assump ion
5.7 is also ul illed.
84 Chap e 5. S a e-space k iging o non-au onomous sys ems
Once eached his poin , i is possible o w i e he ou pu acking e o e m in
ma ix o m. Now, making
L=
Inu0 0 −Inu
.
.
.....
.
..
.
.
0 0 Inu−Inu
,Q=
Q0 0
0...0
0 0 Q
,R=
R0 0
0...0
0 0 R
,
we ob ain
Np−1
X
i=0
ls(˘yi|k,˘uk+i)=(y−ys)⊤Q(y−ys) + u⊤L⊤RLu
= (Mu +n−WYu+ Y)⊤Q(Mu +n−WYu+ Y) + u⊤L⊤RLu.
Also, lo(ys, ) can be w i en as
lo(ys, ) = Y+Wu+Y+ − ⊤OY+Wu+Y+ − .
I only emains o ob ain a ma ix exp ession o l (λ∗
Np|k, λ∗
s). As i was al eady
shown how o ob ain λ∗
s, i is only needed o show how o ob ain λ∗
Np|k. Using
equa ion (5.15), i is clea ha
λ∗
Np|k=Gu+h,
whe e
G=ANp−1B ANp−2B··· B0,
h=ANp−1λ∗
k+
Np−1
X
i=0
ANp−1−ic,
and hus
l (λ∗
Np|k, λ∗
s) = γ(Gu+h−Wu− )⊤P(Gu+h−Wu− ).
I is easy o see ha he op imiza ion p oblem in (5.12) becomes
min
u
1
2u⊤Hu+ ⊤u+ cn
s. . ΨMu ≤∆−Ψn
u≤u≤u.
whe e
H= 2((M−WY)⊤Q(M−WY) + L⊤RL +W⊤Y+⊤OY +W
+γ((G−W)⊤P(G−W))),
⊤= 2(n− Y)⊤Q(M−WY) + 2(Y+ − )⊤OY +W+ 2γ(h− )⊤P(G−W),
cn = (n− Y)⊤Q(n− Y)+(Y+ − )⊤O(Y+ − ) + γ(h− )⊤P(h− ).
5.5. Conclusions 85
Thus, he con ol ac ion is compu ed as
u∗= a g min
u
1
2u⊤Hu+ ⊤u
s. . ΨMu ≤∆−Ψn
u≤u≤u,
whe e only he i s componen o he minimize sequence u∗is applied ollowing
a eceding ho izon scheme as usual in any p edic i e con olle .
86 Chap e 5. S a e-space k iging o non-au onomous sys ems
Pa III
P obabilis ically-ce i ied da a
cen e managemen
89
Chap e 6
Bounds on he cons ain
iola ion le el
6.1 In oduc ion
Bounds on he cons ain iola ion le els a e use ul in cons ained con ol p ob-
lems as hey p o ide a measu e on how likely a e he cons ain s o be iola ed
in p ac ice. Being MPC he mos popula con ol echnique using cons ain s, i
will be chosen o he example in his chap e .
Conside a nonlinea sys em which is also a ec ed by noises and dis u bances,
ha is
xk+1 =h(xk, uk, wk)
yk=g(xk, k),
whe e kis he ime ins an , xk∈Rnxis he s a e o he sys em, uk∈Rnuis he
inpu o he sys em, yk∈Rnyis he ou pu o he sys em, wka e dis u bances in
he s a e, kis measu emen noise and h(·) and g(·) a e nonlinea unc ions such
ha h(·) : Rnx×nu→Rnxand g(·) : Rnx→Rny.
In his case, he con ol ac ion compu ed using a de e minis ic MPC law may no
be op imal o e e y possible ealiza ion o wkand k. Also, i is no possible o
gua an ee ha he cons ain s will be sa is ied. Fo ha eason, i may be needed
o eso o obus MPC schemes, which a e known o be o e ly conse a i e in
mos o he cases as i was showcased in he in oduc ion. Ano he possibili y
would be o ely on s a egies based on andomized se ings such as s ochas ic
MPC o chance-cons ained MPC, which may educe his conse a ism. Howe e ,
his may become an in ac able p oblem some imes.
Ins ead, in his chap e , wo me hodologies o bound he cons ain iola ion a e
o a ini e amily o con olle s by means o o line simula ions o he closed-loop
90 Chap e 6. Bounds on he cons ain iola ion le el
sys em a e p esen ed. This ini e amily o con olle s can be compounded o any
con olle , no necessa ily MPC con olle s. The ob ained bounds will allow us
o compa e he p oposed con olle s in o de o ob ain he bes one acco ding o
a p e-speci ied c i e ion ha could weigh pe o mance and cons ain iola ion.
Also, i does no inc ease he compu a ion bu den o he con ol p oblem online.
6.2 Main esul s
Assume ha a ce ain con olle is pa ame e ized by means o a decision ec o
θ∈Θ, whe e Θ is a se compounded o all he possible alues ha he ec o
θcan ake. The dis u bance w ollows a ce ain p obabili y dis ibu ion whose
sample space is W. No assump ions a e made wi h espec o he size o shape
o W. F om his, a unc ion q: Θ ×W → {0,1}is de ined. This will be help ul
o o mula e he p oblems o his sec ion in a gene al se ing. Thus, he unc ion
q(·) is as ollows
q(θ, w) = (0 i θ ul ills some design speci ica ions o w
1 o he wise.
This speci ica ion could be, o example, he sa is ac ion o a ce ain cons ain
subjec o he ealiza ion w. Then, gi en θ, he p obabili y o iola ion o he
a o emen ioned cons ain o any w∈ W can be w i en as
ψ= P ob {q(θ, w) = 1 |θ}.
As i is almos impossible o compu e o measu e his quan i y exac ly, we ely on
he empi ical mean o app oxima e his alue. Conside wias he i- h ealiza ion
o he dis u bance w, and assuming a numbe o Nsamples, hen
ψ=1
N
N
X
i=1
q(θ, wi),
whe e deno es he empi ical mean. Thus, ψwould be he empi ical mean o
ψ o any gi en expe imen . No e ha as ψis also a andom a iable, any se
o expe imen s, ha is, Ndi e en ealiza ions o q(θ, wi) lead o di e en alues
o ψ.
The objec i e o his chap e is o ob ain igh bounds o his empi ical iola ion
a e.
De ini ion 6.1. Failu e. Assuming ha
E(ψ)> ρ + ∆ρ,
whe e E(·) is he expec a ion ope a o , ρ∈[0,1] and ∆ρ > 0. A ailu e is
conside ed o happen when an empi ical iola ion a e o he con olle ψ ul ills
ψ≤ρ.
6.2. Main esul s 91
Tha is, a ailu e happens when he empi ical iola ion a e leads o misleading
esul s, i.e., when gi en a empi ical iola ion a e ψ≤ρ, he eal iola ion a e ψ
inc eases a quan i y la ge han ∆ρ. The ollowing heo ems p o e ha , wi h a
p ope ly chosen ∆ρ, ailu es occu wi h a p obabili y lowe han a small con iden
pa ame e δ. This de ini ion o ailu e is sligh ly di e en han he one p esen ed,
o example, in [158]. Then, he ollowing heo ems summa ize he esul s o his
chap e .
6.2.1 A i s bound on cons ain iola ion a e
Theo em 6.2 (Bound on he empi ical cons ain iola ion le el).Assuming ha
E(ψ)> ρ + ∆ρ ,
wi h ρ∈[0,1] and ∆ρsa is ying
∆ρ≥1
Nlog 1
δ+ 2 ρ
Nlog 1
δ.(6.1)
Then,
P ob ψ≤ρ< δ ,
which implies ha he eal cons ain iola ion le el is bounded by ρ+ ∆ρwi h
p obabili y g ea e han 1 −δ.
P oo . The objec i e is o uppe bound P ob ψ≤ρunde he assump ion ha
E(ψ)> ρ + ∆ρ. Since he numbe o empi ical iola ions g ows wi h E (ψ), i
is clea ha his p obabili y is lowe bounded by he p obabili y when E (ψ) =
ρ+ ∆ρ. Tha is,
P ob ψ≤ρ|E (ψ)> ρ + ∆ρ<P ob ψ≤ρ|E (ψ) = ρ+ ∆ρ.
Once eached his poin , i is easy o no ice ha he p obabili y o a ce ain con-
s ain o be iola ed can be in e p e ed as a Be noulli andom a iable. The e-
o e, he p obabili y o ha ing less han a ce ain numbe o iola ions o some
numbe o ials can be exp essed as he binomial ail.
On he o he hand, conside a andom a iable X ollowing a binomial andom
dis ibu ion B∼(N, p) whe e Nis he numbe o ials and pis he p obabili y o
success. Then, gi en a cons an x, i is ob ained om Che no ’s bound [128, 159]
ha , i x≤p, hen
P ob X
N≤x|EX
N=p≤e−Nφ(ρ,ρ+∆ρ),
whe e
φ(x, p) = xlog x
p+ (1 −x) log 1−x
1−p.
92 Chap e 6. Bounds on he cons ain iola ion le el
The con e gence a e p o ided by his bound is known o be igh om C am´e ’s
heo em o la ge de ia ions when N→ ∞ ([160], chap e 23 in [161]). I is easy
o see ha he abo e Che no ’s bound can be applied o ou p oblem making
X
N=ψand ∆ρ > 0, ob aining
P ob ψ≤ρ|E (ψ) = ρ+ ∆ρ≤e−Nφ(ρ,ρ+∆ρ).
Thus, his uppe -bound can be designed o be lowe han he con idence pa ame e
δby imposing
e−Nφ(ρ,ρ+∆ρ)≤δ .
Taking loga i hms in bo h sides and ea anging he e ms
φ(ρ, ρ + ∆ρ)≥1
Nlog1
δ.(6.2)
As ∆ρis embedded wi hin he unc ion φ(ρ, ρ + ∆ρ), he bound p esen ed by
Okamo o (Lemma 2 in [159]) is applied o ob ain a closed exp ession o ∆ρ,
φ(ρ, ρ + ∆ρ)≥pρ+ ∆ρ−√ρ2.
Thus, he ollowing su icien condi ion o equa ion (6.2) is ob ained
pρ+ ∆ρ−√ρ2≥1
Nlog1
δ.
Fo ∆ρ≥0, his is equi alen o
pρ+ ∆ρ≥√ρ+ 1
Nlog1
δ,
and hus
ρ+ ∆ρ≥ √ρ+ 1
Nlog1
δ!2
.
This is easily ew i en as
∆ρ≥1
Nlog 1
δ+ 2 ρ
Nlog 1
δ.
This comple es he p oo . ■
6.2.2 A di e en bound
A di e en bound will be gi en in he ollowing. This new bound is no gua an eed
o be igh e han he p e ious one bu i can yield be e esul s in some cases.
6.2. Main esul s 93
Theo em 6.3. Suppose ha
E(ψ)> ρ + ∆ρ ,
wi h ρ∈[0,1] and
∆ρ≥log 1
δ+⌊ρN⌋log a
N1−1
a−ρ, ∀a≥1.(6.3)
Then,
P ob ψ≤ρ< δ ,
As in heo em (6.2), his implies ha he eal cons ain iola ion le el is bounded
by ρ+ ∆ρwi h p obabili y g ea e han 1 −δ.
P oo . Assuming ha E(ψ)> ρ + ∆ρ , one could w i e his p obabili y as
P ob ψ≤ρ|E (ψ)> ρ + ∆ρ.
Again, as he numbe o empi ical iola ions g ows wi h E (ψ), i is clea ha his
p obabili y is lowe bounded by he p obabili y when E (ψ) = ρ+ ∆ρ, i.e.
P ob ψ≤ρ|E (ψ)> ρ + ∆ρ<P ob ψ≤ρ|E (ψ) = ρ+ ∆ρ.
Simila ly, as in heo em 6.1, i is easy o no ice ha his p obabili y can be seen as
a Be noulli andom a iable and hus he p obabili y o ha ing less han a ce ain
numbe o iola ions o some numbe o ials can be exp essed as he binomial
ail
B(N, η, m) =
m
X
y=0 N
yηy(1 −η)N−y,
whe e B(N, η, m) ep esen s he mass p obabili y unc ion o he Binomial dis-
ibu ion. F om he de ini ion o ψ, i is easy o see ha ⌊ψN⌋=ψN whe eas
⌊ρN⌋ ≤ ρN. Then,
P ob ψ≤ρ|E (ψ) = ρ+ ∆ρ= P ob ⌊ψN⌋≤⌊ρN⌋|E (ψ) = ρ+ ∆ρ.
Now, i is possible o ind he equi alence be ween his p obabili y and he bino-
mial ail as
P ob ⌊ψN⌋≤⌊ρN⌋|E (ψ) = ρ+ ∆ρ= B(N, ρ + ∆ρ, ⌊ρN⌋).(6.4)
Lemma 1 in [61] shows ha he binomial ail can be bounded by he ollowing
exp ession
B(N, η, m)≤amη
a+ 1 −ηN,∀a≥1.
100 Chap e 7. Ene gy-e icien managemen o da a cen e s
example, he pape [162] p oposes a con ol a chi ec u e whe e bo h he mal and
asks managemen a e aken in o accoun . The da a cen e is posed as a linea
i s -o de con inuous sys em in i s he mal componen and in he compu a ional
pa . Task a i al a es a e conside ed de e minis ic, hus asks a i e a ixed
in e als. A Quali y o Se ice (QoS) cons ain is used o es ablish s abili y limi s
o he sys em. On he o he hand, in [171], se e al con ol policies assuming ha
he maximum empe a u e o he se e s a e so cons ain s a e p oposed. O he
echniques a ailable a e based on wo kload dis ibu ion wi h some he mal awa e
c i e ia [172, 173]. Also, a solu ion o he p oblem unde he assump ion ha he
e-ci cula ion o ho ai is a cons an empe a u e can be ound in [174].
Model P edic i e Con ol [86] has also been applied o his p oblem. In [175], a
scheduling me hodology based on MPC and elec ici y p ices is p oposed in o de
o educe he economic impac . On he o he hand, [176] uni ies he managemen
wi h he cooling scheme, assuming implici ly ha se e s ha e no limi s o be
o e clocked, which is a he un ealis ic. Due o he complexi y o he p oblem, in
[177] i is p o ided an app oxima ion algo i hm ocused on p o iding as e alua-
ions o he complex cons ain s ha ha e o be aken in o accoun . Howe e , he
models appea ing in he li e a u e, unde di e en assump ions, usually e ade he
ac ha he sys em is ac ually a queue model, simpli ying d as ically he p oblem
a he expense o a less ealis ic modeling.
This chap e p oposes an MPC amewo k o he op imiza ion o cold aisle da a
cen e s. The op imiza ion objec i e is o gua an ee a ce ain QoS o he use s and
keep a sui able empe a u e o he se e s while consuming he leas amoun o
ene gy possible. The da a cen e is modelled as a queue sys em whe e he a i ing
asks can be compu ed by mul iple se e s a he same ime. Al hough i mo es
away om he usual M/M/c queue and makes he ea men mo e complica ed, i
allows mo e gene ali y wi hin he model, whe e i is possible o ake in o accoun
some speci ic kind o da a cen e s like ende a ms [178]. Unlike o he wo ks
a ailable in he li e a u e, he ull queue model is used o p edic he e olu ion
o he sys em h ough he ho izon, hus achie ing a mo e ealis ic modeling. In
o de o ackle andom a i al a es and wo kloads, wo di e en s a egies a e
p oposed. Fi s , i is conside ed ha he a i al a es and wo kloads a e de e min-
is ic, ha is, i is assumed ha hey ake always he alue o he expec a ion o
hei espec i e andom a iables. On he o he hand, a scena io-based app oach
whe e a ce ain numbe o ealiza ions o he dis u bances a e d awn is p oposed
(see chap e 1). The QoS is managed by imposing a ha d cons ain on he num-
be o powe ed se e s in he op imiza ion p oblem, hus gua an eeing ha asks
a e execu ed wi hin he p e-speci ied ime limi s. Al hough he e exis e y op i-
mized mixed-in ege sol e s nowadays such as Gu obi [179], hey a e no sui able
o he ela ed op imiza ion p oblem. Fo ha eason, pa icle algo i hms we e
chosen o deal wi h he op imiza ion p oblem [180]. These algo i hms a e highly
pa allelizable and can ob ain impo an speed ups wi h a pa allel implemen a ion
[181].
7.2. Da a cen e desc ip ion 101
Finally, he bounds ob ained in chap e 6 will be used o quan i y he cons ain
iola ion a e o he p oposed app oaches. This makes possible o ob ain a di e -
en cha ac e iza ion o he cons ain iola ion a e o any p oposed con olle ,
which could be help ul o une some pa ame e s o he con olle s like he con-
ol ho izon o he numbe o scena ios o be conside ed wi hin he op imiza ion
p oblem.
7.2 Da a cen e desc ip ion
Da a cen e s a e compounded o he mally-isola ed uni s in which se e acks
a e alloca ed ypically ollowing a Cold Aisle (CA) s uc u e as i can be seen in
igu e 7.1. He e, he ai low is sepa a ed in o wo di e en lows, being he i s a
cold one which eaches e e y se e ha i is blown om below he loo by means
o sui able buil -in ans. Fo his pu pose, a Compu e Room Ai Condi ioning
(CRAC) uni is esponsible o gene a ing his cold low. Then, he cold ai a els
h ough he se e s and ge ing hea ed h oughou his p ocess and, inally, his
ho ai e u ns o he CRAC uni h ough he ceiling.
CRAC
CRAC
Ceiling
RACKRACK
RACK
CA
Uni
RACKRACK
RACK
Floo
Ho ai
Cold ai
Figu e 7.1: Scheme o a cold aisle da a cen e s uc u e.
The CRAC uni is essen ial wi hin he da a cen e sys em because he se e
empe a u es should be kep below ce ain secu i y le els in o de o ensu e he
se e s eliabili y. As almos all powe consumed by se e s is dissipa ed as hea ,
CRAC ope a ion is e y cos ly because o he high numbe o se e s ha ypical
da a cen e has [166]. Thus, any measu e aimed o achie e an e icien manage-
men o he da a cen e will ha e a g ea impac no only in ope a ing cos s and
en i onmen al impac , bu also in he QoS p o ided o he clien s. In he nex
sec ions, a disc e ized model o he da a cen e dynamics wi h an in eg a ion ime
s ep o sis p esen ed. The disc e e ime uni will be deno ed as k.
102 Chap e 7. Ene gy-e icien managemen o da a cen e s
7.2.1 Tasks model
In his sec ion, he queue model o he da a cen e ope a ion is p esen ed [171].
I is assumed ha he da a cen e has Ma ailable se e s and m, he numbe o
boo ed se e s, i is conside ed as an inpu o he sys em. In his way, he numbe
o se e s wo king in a ce ain ime ins an can be uned acco ding o he needs
o he use s (wo kload) and aking in o accoun e iciency and QoS cons ain s.
Fo he whole da a cen e , he e is a queue whe e he asks wai un il hey can
be p ocessed. Tha is, un il he e is an a ailable se e . As i is accep ed in he
li e a u e [171, 176], he ime be ween a i als is assumed o ollow an exponen ial
dis ibu ion wi h mean kaand a p obabili y densi y unc ion
(k) = 1
ka
e−k
ka.(7.1)
Also, le Lkbe he eques a e a ins an k. Tha is, he numbe o asks a i ing
o he da a cen e a ins an k. Because he ime be ween a i als was p e iously
assumed o ollow an exponen ial dis ibu ion, Lkcan be modeled as a Poisson
andom a iable wi h mean 1
kaand p obabili y mass unc ion
g(n) = 1
n!1
kan
e−1
ka.(7.2)
The numbe o asks a i ing in he in e al [k, k −ku] is deno ed as Lku,k o an
in e al o leng h ku>0, ku∈Z. Also, he p obabili y mass unc ion is
gku(n) = 1
n!ku
kan
e−ku
ka,(7.3)
wi h mean ku
ka.
As i holds in p ac ice, i is conside ed ha asks ha e a di e en wo kload. Then,
o a ce ain ask, he compu a ional ime equi ed o comple e i in a single se e
is assumed o ollow an exponen ial dis ibu ion like (7.1) wi h mean 1
µ. Thus,
Wkco esponds o he a e age wo kload o he Lk asks ha a i ed a ins an
k. Simila ly, Wku,k co esponds o he a e age wo kload o he Lku,k asks ha
a i ed be ween kand k−ku. Also, he pa ame e s kaand µde ine he minimum
numbe o se e s ha should be u ned on wi hin he da a cen e so ha he
queue does no g ow in ini ely. Then, Mmus sa is y he condi ion
M≥1
kaµ.(7.4)
7.2.2 Se e model
Fo a ce ain se e i, he s a e xk,(i)a ins an kis conside ed o be compounded
o :
7.2. Da a cen e desc ip ion 103
1. The numbe o asks cu en ly unning in he da a cen e (αk).
2. The numbe o asks in he queue (βk).
3. The empe a u e o he cold ai (Tc,k).
4. The empe a u e o he se e (Tk,(i)).
5. The ime ins an in which he se e iis u ned on (si).
Ob aining he s a e o he whole da a cen e is easy as i su ices o include he
emaining Tk,(i)and s(i). The e o e, x⊤
kis ob ained as
x⊤
k=αk, βk, Tc,k, Tk,(1) . . . Tk,(M), s(1) . . . s(M).
On he o he hand, se e s swi ch be ween ou possible wo king condi ions:
•O . The se e does no d aw any powe .
•Boo ing. This wo king condi ion appea s due o he ixed delay kon ha i
akes o u n on a se e om o o wo king o idle. While in his ansi ion
ime, he se e d aws powe a he same a e as he idle condi ion.
•Wo king. The se e is “on” and i is p ocessing a ce ain ask. Fo sim-
plici y easons, i is assumed ha he powe is d awn a a cons an a e. In
p ac ice, his is equi alen o assume ha he CPU equency is cons an .
•Idle. The se e is “on” bu i is no p ocessing any asks, d awing less
powe han in he wo king s a e. Howe e , he se e is consuming ene gy
o doing no hing.
As usual in he li e a u e and o simplici y easons, he ansi ion om “on” o
“o ” is assumed o be ins an aneous. Also, he ansi ion om wo king o idle
and ice- e sa is conside ed negligible. No e ha because o he ansi ion ime
kon and he powe d awn h oughou he boo ing p ocess, i may be ad an ageous
o keep a ce ain numbe o se e s in idle s a e in he case when hey a e expec ed
o be needed in a nea u u e. Fu he mo e, he ansi ion om “o ” o “on” is
always conside ed o be immedia ely a ailable (al hough i akes a kon ime o be
comple ed), bu he e e se, ha is, om “on” o “o ”, is done in a de e ed way,
because he se e mus inish he emaining asks.
Taken in o accoun he p e ious wo king condi ions, le uk,(i)be an inpu ha
indica es i he se e iis swi ched on (uk,(i)= 1) o o (uk,(i)= 0) a a ce ain
ime ins an k. Then, he powe consump ion o a se e iis de ined by he
ollowing condi ions
pk,(i)(xk,(i), uk,(i)) =
0 i o (uk,(i)= 0)
a2i boo ing (uk,(i)= 1, αk≥0, k −si< kon)
a2i idle (uk,(i)= 1, αk= 0, k −si≥kon)
a1+a2i wo king (uk,(i)= 1, αk>0, k −si≥kon)
104 Chap e 7. Ene gy-e icien managemen o da a cen e s
whe e a1is he ma ginal consump ion and a2 he minimum consump ion.
The managemen app oach p esen ed in his hesis does no deal wi h he ask
scheduling o he da a cen e [182]. Fo ha eason, i is assumed ha he dis-
ibu ion o he asks among he se e s ollows some known ules. I is also
assumed ha a ask can be spli among mul iple se e s (up o M). Howe e , a
single se e can wo k only o he comple ion o a single ask. Because o he p e-
ious assump ion, he da a cen e will no wo k as a M/M/c queue in which each
ask is scheduled o be execu ed in a single se e , complica ing he ma hema ical
modelling. Howe e , i will esul in a mo e gene al da a cen e model.
These se e assignmen policies imply he exis ence o a pool o unning asks
whose leng h is equal o he numbe o se e s in “on” condi ion (i.e., “wo king”
o “idle”). Once a ask is eady o be p ocessed, ha is, i is a he on o he
queue and he pool has a leas one emp y slo , i is assigned o a leas one se e
and i ne e lea es he pool un il i s comple ion. Thus, he emaining wo kload
o each ask is s ic ly dec easing.
Deno ing mkas he numbe o se e s u ned on a ins an k(i.e. wo king o
idle), he p ocessing o a ask is done in he ollowing way. Assume ha only a
ce ain ask is wi hin a pool o mkse e s (αk= 1). The wo kload o he ask is
he numbe o “wo k packages” ha need o be p ocessed o comple e he ask.
A ins an k,mkse e s a e assigned o his ask (because i is he only ask
in he pool). Thus, e e y se e will compu e a “wo k package” esul ing in mk
“wo k packages” execu ed. I he e a e no “wo k packages” le o he ins an
k+ 1, he ask is comple ed and ejec ed om he pool. I he pool is ull a a
ce ain ins an k(i.e. αk=mk), he assigna ion is i ial because e e y ask can
only be assigned one se e . Fo he case whe e 1 < αk< mk, he asks will be
assigned o one se e and he ask wi h la ges emaining ime will be assigned
o mk−αkse e s. This makes sense because i is easy o see ha i will achie e
be e QoS.
7.2.3 The mal model
The he mal model o he se e s is de i ed om he he mal balance equa ions
K
dT(i)( )
d =cpqa( )(Tc( )−T(i)( )) + p(i)( ),(7.5)
whe e Tcand qaa e he empe a u e and low o he cold ai p o ided by he
CRAC uni , K he se e he mal capaci y, T(i)and p(i)a e he empe a u e and
powe consump ion o se e i espec i ely and cp he ai hea capaci y. Con-
side ing a disc e iza ion scheme wi h an in eg a ion s ep s, equa ion (7.5) u ns
in o
Tk+1,(i)=Tk,(i)+ s
K cpqa,k Tc,k −Tk,(i)+pk,(i).(7.6)
7.2. Da a cen e desc ip ion 105
A ha d cons ain is used o gua an ee he se e eliabili y, keeping he empe -
a u es unde ce ain sa e y le els. Tha is
T(i)≤80◦C∀i∈M .
F om he a iables a ec ing T(i)in (7.6), he low qa,k is assumed o be cons an
and only p(i)and Tccan be conside ed manipulable, he i s one h ough he
s a e o he se e and he second one is assumed o be egula ed by a se poin
T . Tha is, Tc ollows T wi h a i s o de closed loop dynamics wi h a ime
cons an τand uni y gain
τdTc( )
d =T ( )−Tc( ).(7.7)
Simila ly, equa ion (7.7) u ns in o
Tc,k+1 =Tc,k + s
τ(T ,k −Tc,k).(7.8)
On he o he hand, ha d cons ain s in he inpu T a e conside ed
15◦C≤T ≤25◦C.
Also, he coe icien o pe omance (CoP) o he CRAC uni will change depend-
ing on he cold ai empe a u e (Tc). The CoP ep esen s how expensi e is he
cooling p ocess o he ai low un il a ce ain empe a u e. Usually, mo e powe
consump ion is equi ed o each lowe empe a u es. As he CoP inc eases, he
cos will dec ease. I can be calcula ed om he ollowing equa ion
CoP(Tc,k) = 0.0068 T2
c,k + 0.0008 Tc,k + 0.458 ,(7.9)
which is widely adop ed in he li e a u e [173]. Thus, le
M
P
i=1
pk,(i)(xk,(i), uk,(i)) be
he se e powe consump ion a ime ins an k. The powe consump ion a he
CRAC uni can be compu ed as
M
P
i=1
pk,(i)(xk,(i), uk,(i))
CoP(Tc,k).
Thus, he o al powe consump ion co esponds o he powe d awn by he se e s
added o he CRAC powe consump ion, leading o
M
X
i=1 1 + 1
CoP(Tc,k)pk,(i)(xk,(i), uk,(i)).
106 Chap e 7. Ene gy-e icien managemen o da a cen e s
7.2.4 Quali y o se ice
The QoS o a ask is de ined as he ime equi ed o inish i since i s a i al un il
i s comple ion. I includes bo h he wai ing ime wi hin queue and he execu ion
ime once i is in he pool. Gua an eeing ha his ime will be lowe han an
ag eed one is a necessa y ope a ing condi ion.
In he case whe e he asks a e assigned o jus one se e , like in an M/M/c
queue (o E lang-C model [183]), he mean se ice ime ollows he exp ession
c,k =1
mk
Wk−Lk
.(7.10)
This measu emen could be used in p ac ice wi h es ima ions o Wkand Lk,
deno ed as ˜
Wkand ˜
Lk espec i ely. Howe e , in he p oposed app oach, as a
ce ain ask can be execu ed concu en ly in many se e s, he a o emen ioned
measu e is used o p o ide an uppe bound and hus, in his case, i can only be
lowe o equal
s ,k ≤1
mk
˜
Wk−˜
Lk
,
whe e ˜
Wkis he es ima ed alue o Wka ins an k o he nex ˜
Lk eques s and
˜
Lkis he es ima ion o Lka ins an k. I is conside ed ha he QoS is sa is ied
a ins an ki he mean se ice ime is no la ge han a speci ied alue D. In
o he wo ds, he QoS cons ain is sa is ied i
1
mk
˜
Wk−˜
Lk≤D,
which implies ha
mk≥˜
Wk1
D+˜
Lk.
The e m on he igh ep esen s he minimum numbe o se e s o ul ill he
QoS cons ain wi h a ce ain ˜
L,˜
Wand D. This can be w i en as
mD,k =˜
Wk1
D+˜
Lk.
7.3 Managemen app oach
In his chap e , an op imal managemen policy inspi ed on p edic i e con ol
s a egies is p oposed. This con olle decides he numbe o se e s ha should
be on o o and he se poin empe a u e o he CRAC uni . The op imiza ion
objec i e used o decide he op imal alues o he inpu s will be he minimiza ion
o he ene gy consump ion and he con ol e o o he inpu s (i.e. se e swi ching
and empe a u e e e ence changes), subjec o he mal and QoS cons ain s.
7.3. Managemen app oach 107
Fi s o all, i is assumed ha he dynamics o he queue, ask a i als, e c. un
much as e han he p ocess o swi ching on a se e (i.e. kon ≫1). Ha ing
such a la ge delay in he con ol ac ions, he con ol decisions ha e o be made
in a supe io ime scale and sepa a e hem o e ime. This leads us o a scheme
whe e he p edic i e con olle is no execu ed a e e y ins an kbu i is execu ed
a e e y ins an km swhe e kmis he numbe o ins an s be ween he con olle
execu ion. Thus, he sample ime o he con olle is km s. In o de o a oid
swi ching o a se e ha is s ill boo ing ( ha is, ne e eached he “on” s a e
and did no hing bu consuming powe ), a sample ime la ge han he swi ching
on ime is chosen, i.e. km≥kon.
Based on he model, he expec ed e olu ion o he da a cen e can be es ima ed
and hen, i is possible o associa e a p edic ed cos o a sequence o candida e
u u e con ol inpu s. Fo cla i y pu poses, deno e ℓj=k+jkm. This will wo k as
he ime scale o he MPC con olle . In his chap e , he ollowing p edic ed cos
unc ion o measu e he expec ed pe o mance o he da a cen e will be used:
V(ℓ0,x,u,T ) =
Np
X
j=0
M
X
i=0 1 + 1
CoP(Tc,ℓj)pℓj,(i)(xℓj,(i), uℓj,(i))
+κu
Nc
X
j=0|∆uℓj|+κT
Nc
X
j=0|∆T ,ℓj|,(7.11)
whe e Npand Nca e he p edic ion and con ol ho izons, xis he sequence o
xℓjo e he p edic ion ho izon, uand T a e he sequences o “on”-“o ” con ol
ac ions o all se e s and empe a u e se poin s h ough he con ol ho izon
espec i ely, ∆uℓjis he o al numbe o commu a ions o ei he “on” o “o ”
a ins an ℓj, ∆T ,ℓjis he inc emen in T ,ℓj,κuis a e m weigh ing ∆uℓjand
κT is a e m weigh ing ∆T ,ℓj. Also, he con ol ac ions u he om he con ol
ho izon a e conside ed o emain cons an .
In o de o de i e he p oposed con olle , i is necessa y o de e mine a p edic ion
model such ha o a gi en s a e a ime ℓj,xℓjand o gi en con ol ac ions
uℓjand T ,ℓj( ha will emain cons an h oughou he sampling ime km), he
s a e o he da a cen e p edic ed a he nex sampling ime ˜xℓj+1 is calcula ed
depending on he es ima ion o he numbe o asks and hei wo kload ˜
Lkm,ℓj+1
and ˜
Wkm,ℓj+1 . This p edic ion model can be posed as:
˜xℓj+1 =h(xℓj, uℓj, T ,ℓj,˜
Lkm,ℓj+1 ,˜
Wkm,ℓj+1),
being h(·) he unc ion ha compu e he ollowing s a e gi en he p e ious one,
he inpu s and he ealisa ions o Lkmand Wkm. No e ha he unc ion h(·)
mus compu e all e en s happening ho ough he in e al km o e e y ins an k
in o de o be able o e u n he s a e a he ollowing sample ime. In p ac ice,
his unc ion is e alua ed by means o an open-loop simula ion o he comple e
queue model o he da a cen e .
108 Chap e 7. Ene gy-e icien managemen o da a cen e s
The op imal p edic ed numbe o se e s and se poin empe a u es o he CRAC
will be hen compu ed as he solu ion o he op imiza ion p oblem
min
mℓj,T ,ℓj
V(ℓ0,x,u,T ) (7.12a)
s. . ˜xℓj+1 =h(xℓj, uℓj, T ,ℓj,˜
Lkm,ℓj+1 ,˜
Wkm,ℓj+1 ) (7.12b)
mℓj∈[1, M]∀j∈[0, Nc] (7.12c)
uℓj= swi ch(mℓj)∀j∈[0, Nc] (7.12d)
15◦C≤T ,ℓj≤25◦C∀j∈[0, Nc] (7.12e)
Tℓj,(i)≤80◦C∀i∈M∀j∈[1, Np] (7.12 )
mℓj≥mD,ℓj∀j∈[1, Np],(7.12g)
whe e swi ch(·) ep esen s he policy o selec which se e s a e o be swi ched on.
He e, a simple scheme is p oposed o such policy. Tha is, o a gi en numbe o
se e s, he i s mkwill be swi ched on.
As i is cus oma y in p edic i e con olle s, he solu ion o (7.12) is applied in a
eceding ho izon manne , meaning ha only he i s con ol ac ions and empe -
a u e se poin s a e ac ually applied (i.e., uℓ0,(i)and T ,ℓ0) while he emaining
decision a iables (uℓ1,(i), . . . , uℓNc−1,(i)and T ,ℓ1, . . . , T ,ℓNc−1) compu ed a ime k
a e disca ded. The op imiza ion o (7.12) is hen epea ed a each sampling ime
so ha he decision a iables o be applied a e compu ed using he eal s a e o
he da a cen e a ha sampling ime.
I should be no ed ha in his op imiza ion p oblem some o he a iables a e
in ege (numbe o se e s on) whe eas o he a e eal alued ( he se -poin empe -
a u es). This, oge he wi h he complexi y o he da a cen e model mo i a es
he use o specialized op imiza ion algo i hms o sol e (7.12), such as a pa i-
cle based op imiza ion echnique [180, 181] ha will be exposed in he ollowing
sec ion.
7.4 Pa icle based sol e s o complex op imiza ion
p oblems
In hese echniques o i e a i e na u e, a se o possible candida e solu ions (called
pa icles) a e e alua ed a each i e a ion and used o gene a e a new candida e
solu ion se ha may be close o he solu ion o he op imiza ion p oblem. He e,
he e alua ion o each pa icle will be done by means o simula ions ha will be
used o assess he pe o mance o each candida e solu ion. The main ing edien s
o he p oposed echnique a e:
•Pa icles a e candida e solu ions o he op imiza ion p oblem. Tha is, a
sequence o con ol ac ions o e he con ol ho izon. A highe p oblem
7.4. Pa icle based sol e s o complex op imiza ion p oblems 109
complexi y (i.e., mo e se e s and longe con ol ho izons), mo e pa icles
a e needed in o de o ob ain a solu ion close o he op imal one [184].
•Weigh s. Pa icles ha e associa ed a weigh ha ep esen s how good is
a pa icle compa ed wi h he o he s. In his wo k, he weigh is based
on he pe o mance cos o each pa icle compu ed by means o a compu e
simula ion o he da a cen e model. These simula ions p edic he e olu ion
o he da a cen e along he p edic ion ho izon i he decision a iables a e
hose o he pa icles. Once he simula ions o all pa icles a e comple ed
( his can be done in pa allel), a scaling o he pe o mance cos s o he
pa icles is made. Le Vmax and Vmin be he maximum and minimum cos s
a ained on he se o pa icles unde e alua ion. I a gi en pa icle z≜
(m,T ) has a pe o mance cos V(z), hen he weigh o he pa icle zwill
be
σ(z)= 1 −V(z)−Vmin
Vmax −Vmin
,(7.13)
meaning ha pa icles wi h highe cos s will ha e lowe weigh s and ice
e sa. This is e y impo an o he esampling phase.
•Feasibili y checking. Those pa icles ha do no sa is y he cons ain s in
(7.12) will be assigned a ze o weigh so ha hey canno be selec ed in he
esampling s age.
•Resampling is he s ep whe e a new gene a ion o pa icles is c ea ed based
on he pe o mance (weigh s) a he p e ious i e a ion. This is done by
means o he Ki agawa esampling algo i hm wi h s a i ied scheme [185].
The aim o his me hod is o o m g oups o pa icles wi h good pe o mance
in di e en a eas o he easible se o solu ions. In his way, he isk o
ge ing s uck a a local minimum is educed because he algo i hm conside s
all pa icles and no only he bes one. Pa icles wi h highe weigh s a e
mo e likely o be esampled a nex i e a ion (i.e., selec ed o be included
in he nex pa icle se ).
•Pe u ba ion. This p ocess is inhe en ly connec ed o he p e ious one. Once
pa icles a e esampled, i is necessa y o ”mo e” hem along he local sea ch
space in o de o disco e new easible solu ions wi h po en ially lowe cos s.
In his wo k, he pe u ba ion is made by adding a Gaussian whi e noise.
Figu e 7.2 shows he main ideas o he esampling and pe u ba ion s eps
in a 2 deg ee o eedom minimiza ion example. A i s , he e exis s a se
o andom-gene a ed pa icles. As he i e a ions con inue, pa icles mo e
owa ds be e possible solu ions acco ding o he cos s ob ained p e iously.
In ou case, he op imiza ion a iables a e in ege , hus his p ocess ends
ounding o he nea es in ege .
•Reseeding. The ini ial se o pa icles should be gene a ed in a andom
manne . Howe e , o he subsequen sample imes, he eseeding can be
116 Chap e 7. Ene gy-e icien managemen o da a cen e s
0 500 1000 1500 2000 2500
0
0.5
1
1.5
0 500 1000 1500 2000 2500
4
6
8
10
0 500 1000 1500 2000 2500
Time (s)
0
2
4
6
Figu e 7.3: F om op o bo om, Mean Queue Leng h, Mean Numbe o On Se e s and
Mean Numbe o Idle Se e s
con olle . The op subplo shows he powe consump ion e m in Wa s (W). As
i can be guessed, he con olle s wi h la ge numbe o ac i e se e s will ha e
g ea e powe consump ion, due o he consump ion o he se e s hemsel es
bu also o he g ea e consump ion o he CRAC uni (which ollowing (7.9) is
assumed p opo ional o ha o he se e s).
The lowe subplo s a e e e ed o he con ol e o s o he con ol ac ions. He e,
i can be seen ha he con olle s wi h Nc= 5 end o change mo e hei con ol
ac ions which leads o g ea e con ol e o s, bu also acili a es a mo e uned
applica ion o he con ol ac ion ha esul s in lowe o e all cos s as seen in Figu e
7.5. In his igu e, he ins an aneous o al cos o each con olle is shown. As
expec ed, he mo e conse a i e con olle s a e he mos expensi e ones in e ms
o pe o mance cos .
7.6.1 QoS iola ion a e
Once all simula ions a e ca ied ou , he bounds can be compu ed o each one
o he ou con olle s. Table 7.2 shows he esul s o he QoS cons ain . The
inal esul is shown in he ρ+ ∆ρcolumns which a e he uppe bounds (wi h
con idence 1 −δ) o he p obabili y o no mee ing he ag eed se ice ime D o
he heo ems 2.2 and 2.3 espec i ely. Lowe numbe s imply be e p obabilis ic
7.6. Nume ical esul s 117
0 500 1000 1500 2000 2500
1500
2000
2500
3000
3500
0 500 1000 1500 2000 2500
0
500
1000
0 500 1000 1500 2000 2500
Time (s)
0
200
400
Figu e 7.4: F om op o bo om: Mean Powe Consump ion, Mean Con ol E o o
boo ing se e s and Mean Con ol E o o changing he CRAC empe a u e e e ence
gua an ees, which as expec ed co esponds o he mo e conse a i e con olle s
(C1and C2).
ρ ρ + ∆ρ(2.2) ρ+ ∆ρ(2.3) N
C1S-Pbo, Nc= 1 0.0032 0.0050 0.0045 65960
C2S-Pbo, Nc= 5 0.0040 0.0060 0.0054 65926
C3Pbo, Nc= 1 0.0278 0.0328 0.0314 65708
C4Pbo, Nc= 5 0.0337 0.0392 0.0376 65678
Table 7.2: Viola ion a e o he p oposed con olle s o he QoS cons ain s wi h δ= 10−6
7.6.2 The mal cons ain iola ion a e
Bounds ha e also been ob ained o he empe a u e cons ain , wi h able 7.3
summa izing he esul s. In his case, Ns ands as he numbe o o al ime
ins an s o all he simula ions H(which we e all o he same leng h), and he
uppe bound is on he p obabili y o eaching a empe a u e highe han 80◦C.
Again he mo e conse a i e a con olle has he be e p obabilis ic gua an ees,
bu a he expense o a highe cos .
118 Chap e 7. Ene gy-e icien managemen o da a cen e s
0 500 1000 1500 2000 2500
Time (s)
1800
2000
2200
2400
2600
2800
3000
3200
3400
3600
Figu e 7.5: Ins an aneous o al cos
ρ ρ + ∆ρ(2.2) ρ+ ∆ρ(2.3) N
C1S-Pbo, Nc= 1 0.0000 0.0055 0.0058 2500
C2S-Pbo, Nc= 5 0.0000 0.0055 0.0058 2500
C3Pbo, Nc= 1 0.0001 0.0071 0.0067 2500
C4Pbo, Nc= 5 0.0000 0.0055 0.0058 2500
Table 7.3: Viola ion a e o he p oposed con olle s o he maximum empe a u e con-
s ain wi h δ= 10−6
7.6.3 Pa allel compu a ion imp o emen
In o de o p o e he speed-ups ob ained wi h a pa allel implemen a ion, he
algo i hms ha e been implemen ed in he CPU and also in a CUDA capable
GPU. The CPU e sion is coded comple ely in Ma lab while he GPU e sion
uses Ma lab code o he se ial ope a ions and C code o he CUDA ke nels.
Table 7.4 shows he execu ion ime o he con olle s o an inc easing numbe o
pa icles. Cells wi h a hyphen mean ha he ime exceeded he sampling ime.
Finally, igu e 7.6 shows he ela i e speed-up o di e en numbe s o pa icles.
S-Pbo GPU S-Pbo CPU Pbo GPU Pbo CPU
10 17.0931 23.4282 0.6974 1.0001
100 17.8046 - 0.7230 9.4192
1000 30.2881 - 1.2267 93.6546
10000 91.4658 - 3.7208 -
Table 7.4: Mean compu a ion ime o he con olle s in seconds.
7.6. Nume ical esul s 119
0 100 200 300 400 500 600 700 800 900 1000
Numbe o Pa icles ( )
0
10
20
30
40
50
60
70
80
Execu ion speed-up
Figu e 7.6: Speed-up ob ained wi h he GPU compu ing o he con olle execu ion.
As i can be expec ed, a a highe numbe o pa icles, he execu ion ime wi hin
he CPU g ows s ongly whe eas he compu a ion ime wi hin he GPU yields
almos cons an in compa ison, which leads o he inc easing slope shown in igu e
7.6. I also should be no ed ha once eached a numbe o pa icles g ea e han
1000, he execu ion imes o he CPU become unmanageable and he simula ions
a e ex emely cos ly.
7.6.4 Compu a ion ime analysis
Also, an analysis o he compu a ion ime o he algo i hm wi h espec o di e en
numbe o se e s has been done in o de o s udy i s beha io o he p oblem
o la ge da a cen e s. The esul s a e shown in igu e 7.7. A polynomial o o de
n= 2 is also added o he igu es, con i ming a quad a ic complexi y.
0 50 100 150 200 250 300
Numbe o Se e s
0
50
100
150
200
250
Compu a ion Time (s)
Real
Fi ing
25 30 35 40 45 50 55 60
Numbe o Se e s
20
40
60
80
100
120
140
160
180
200
Compu a ion Time (s)
Real
Fi ing
Figu e 7.7: Compu a ion imes o he algo i hms o an inc easing numbe o se e s.
Le : Algo i hm wi hou scena ios. Righ : Algo i hm wi h scena ios
Thus, he algo i hm is easible om a compu a ional poin o iew and i can be
120 Chap e 7. Ene gy-e icien managemen o da a cen e s
used wi h la ge da a cen e s. Ne e heless, o highe numbe s o se e s, besides
adding mo e compu a ional powe , one could conside mul iple con olle s each
one o each cold aisle o conside ha he con ol ac ions handle clus e s o se e s
ins ead o indi idual ones.
7.7 Conclusions
This chap e p esen ed a complex model o a da a cen e and p oposed a me hodol-
ogy o deal wi h he p oblem o minimizing he powe consump ion o he acili ies
while main aining he quali y o se ice a accep able le els. The p oposed MPC-
inspi ed con olle simula es he whole da a cen e model in o de o compu e
he op imal inpu s o be applied o he sys em. Fo his eason, a pa icle-based
algo i hm was used o sol e he op imiza ion p oblem a each ime ins an k.
The bounds p oposed in chap e 6 we e used o une he hype pa ame e s o he
con olle s.
121
Chap e 8
Conclusions and u u e wo k
This chap e b ie ly summa izes he con ibu ions o his disse a ion and p esen s
some u u e lines o wo k in each ield.
8.1 Con ibu ions
As i was s a ed in he in oduc ion, he con ibu ions o his hesis a e ela ed
o h ee di e en ields: sys em iden i ica ion, p obabilis ic o ecas ing and model
p edic i e con ol. In wha ollows, he con ibu ions o each chap e a e s a ed:
•Chap e 2 p esen ed he no ion o dissimila i y and he dissimila i y unc ion
o be used h oughou his disse a ion. I was shown ha he p oposed
dissimila i y unc ion has many ad an ageous p ope ies. Also, by means o
some o ecas ing examples, i was shown ha he dissimila i y unc ion can
be used as a p edic o , ob aining be e esul s han hose ob ained wi h a
neu al ne wo k in he ield o s ock o ecas ing.
•In chap e 3, he case o unce ain y p edic ions was ackled. Ins ead o
p o iding jus he expec ed alue o he o ecas ing, a p edic ion egion is
ob ained. Fi s , a me hod o ien ed o ob ain in e al p edic ions o a uni a i-
a e sys em o ime se ies was p oposed. This me hod was compu a ionally
expensi e because i equi ed he in eg a ion o he empi ical condi ioned
p obabili y densi y unc ion o e e y ime ins an . In any case, he nu-
me ical examples showed ha he pe o mance was be e in compa ison o
se -membe ship and quan ile eg ession me hods.
Then, a me hod o bypass he nume ical in eg a ion o he empi ical con-
di ioned p obabili y densi y unc ion was p oposed. Besides his g ea ad-
an age, which educes conside ably he execu ion ime o he algo i hm,
mul i a ia e p edic ion egions could be ob ained. Also, i was shown ha
ellipsoidal app oxima ions o such egions can be easily ob ained. Finally,
122 Chap e 8. Conclusions and u u e wo k
by means o a nume ical example, i was shown ha he p oposed app oach
and he ellipsoidal app oxima ions a e smalle han hose egions ob ained
wi h quan ile eg ession, gaussian p ocesses o in e se eg ession me hods
while ul illing he p obabilis ic speci ica ions.
•Chap e 4 p esen ed he s a e-space k iging me hod o au onomous sys-
ems. By using his echnique, a model o he sys em can be ob ained using
only his o ical da a o he p ocess. Two di e en a ia ions a e p oposed,
one ob ained by weigh ing he locali y wi hin he dissimila i y unc ion and
ano he one whe e he nonlinea i y is modeled by means o ke nel unc-
ions. Also, i was shown ha he kalman il e can be used o imp o e he
quali y o he p edic ions. Finally, wo nume ical o ecas ing examples we e
p esen ed, whe e i was shown ha he p oposed app oaches ob ain be e
esul s han o he machine lea ning echniques.
•On he o he hand, chap e 5 p esen ed he s a e-space k iging me hod
o non-au onomous sys ems. Thanks o he modi ica ions made o he
cons ain s in he dissimila i y unc ion, i is possible o include he e ec
o an ex e nal inpu . As he models ob ained in his chap e a e sui able o
MPC, a MPC o mula ion using he K-SSK model was p oposed. Finally, a
simula ion example and a eal expe imen we e conduc ed in o de o show
he good pe o mance o he con olle .
•In chap e 6, by means o he Che no ’s bound [128, 159] and he esul s by
Alamo e al. in [61], wo di e en sha p bounds o he cons ain iola ion
a e o a ce ain con olle we e ob ained. No assump ions on he p obabili y
dis ibu ions o he dis u bances we e made in o de o p o e he ob ained
esul s. I is only equi ed o ob ain i.i.d. samples, which is a s anda d
assump ion in he ield. Finally, a simple MPC example was p oposed o
show how he p oposed bounds should be compu ed.
•Chap e 7 showcased a mo e complica ed example whe e he p oposed bounds
can be used. The chosen sys em was a da a cen e . This kind o acili ies
ha e an ex emely la ge elec ical consump ion due o bo h IT equipmen
and in as uc u e acili ies. To ackle his p oblem, an ene gy-e icien man-
agemen app oach was p oposed. He e, he bounds p oposed in chap e 6
we e used o une he pa ame e s o he con olle .
8.2 Fu u e wo k
The u u e lines o wo k o each one o he pa s o his disse a ion can be ound
in he ollowing:
•As he ob ained in e al p edic ions and p edic ion egions o pa I pe -
o med good enough in a o ecas ing example, i would be in e es ing o
apply hese unce ain p edic ions in a con ol scheme, i.e. de eloping new
8.2. Fu u e wo k 123
s ochas ic MPC con olle s whose p edic ion model is gi en by he p edic-
o s ob ained in chap e 3.
The mos app oachable way would be o syn hesize a con olle o a uni-
dimensional sys em whe e he p oposed in e al p edic o is used o ob ain
unce ain p edic ions o he sys em. These p edic ions would be used wi hin
he op imiza ion p oblem o ensu e he cons ain sa is ac ion in a chance-
cons ained manne .
•The p oposed MPC o mula ion in pa II p esen ed nominal s abili y and
obus asymp o ic s abili y, bu lacked obus cons ain sa is ac ion. Fo
his eason, i would be in e es ing o imp o e he con olle o mula ion
o ackle his case. Also, he in eg a ion o a eal ime op imize and he
conside a ion o economic objec i es in he con olle is ano he pending
ask.
On he o he hand, he p oposed SSK models ha dly accep new da a be-
cause his would mean ha he s a e o he sys em is cons an ly inc easing.
I is necessa y o ind a good me hod ha would allow he use o in oduce
new da a o he SSK model wi hou inc easing he complexi y.
•When some o he p oposed bounds in pa III a e used, i is possible o lose
he ecu si e easibili y o he MPC con olle unless he con olle o mu-
la ion is elaxed somehow. Fo ha eason, a solu ion o his phenomenon
should be p oposed a e a ca e ul analysis o i s o igin.
On he o he hand, he i.i.d. assump ion, al hough usual in he li e a u e,
is ha d o be ul illed because samples d awn om a dynamical sys em a e
inhe en ly co ela ed. I would be in e es ing o de elop me hods which
allow us o ob ain samples ha a e as i.i.d. as possible.
124 Chap e 8. Conclusions and u u e wo k
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