IFAC Pape sOnLine 53-2 (2020) 11271–11276
ScienceDi ec
ScienceDi ec
A ailable online a www.sciencedi ec .com
2405-8963 Copy igh © 2020 The Au ho s. This is an open access a icle unde he CC BY-NC-ND license
.
Pee e iew unde esponsibili y o In e na ional Fede a ion o Au oma ic Con ol.
10.1016/j.i acol.2020.12.362
10.1016/j.i acol.2020.12.362 2405-8963
Copy igh ©
2020 The Au ho s. This is an open access a icle unde he CC BY-NC-ND license
(
h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0
)
A p obabilis ic alida ion app oach o
penal y unc ion design in
S ochas ic Model P edic i e Con ol
Ma ina Mamma ella ∗1Teodo o Alamo ∗∗ Se gio Lucia ∗∗∗
Fab izio Dabbene ∗
∗Ins i u e o Elec onics, Compu e and Telecommunica ion
Enginee ing, Na ional Resea ch Council o I aly, Tu in, I aly (e-mail:
ma ina.mamma ella@ieii .cn .i , ab izio.dabbene@ ieii .cn .i ).
∗∗ Depa amen o de Ingenie ´ıa de Sis emas y Au om´a ica, Uni e sidad
de Se illa, Escuela Supe io de Ingenie os, Camino de los
Descub imien os s/n, 41092 Se illa, Spain (e-mail: [email protected])
∗∗∗ Eins ein Cen e Digi al Fu u e, Technische Uni e si ¨a Be lin,
Ge many (e-mail: se gio.lucia@ u-be lin.de)
Abs ac : In his pape , we conside a s ochas ic Model P edic i e Con ol able o accoun o
e ec s o addi i e s ochas ic dis u bance wi h unbounded suppo , and equi ing no es ic i e
assump ion on ei he independence no Gaussiani y. We e isi he a he classical app oach
based on penal y unc ions, wi h he aim o designing a con ol scheme ha mee s some
gi en p obabilis ic speci ica ions. The main di e ence wi h p e ious app oaches is ha we do
no ecu o he no ion o p obabilis ic ecu si e easibili y, and hence we do no conside
sepa a ely he un easible case. In pa icula , wo p obabilis ic design p oblems a e en isioned.
The i s andomiza ion p oblem aims o design o line he cons ain se igh ening, ollowing
an app oach inhe i ed om ube-based MPC. Fo he second p obabilis ic scheme, a speci ic
p obabilis ic alida ion app oach is exploi ed o uning he penal y pa ame e , o be selec ed
o line among a ini e- amily o possible alues. The simple algo i hm he e p oposed allows
designing a single con olle , always gua an eeing easibili y o he online op imiza ion p oblem.
The p oposed me hod is shown o be mo e compu a ionally ac able han p e ious schemes.
This is due o he ac ha he sample complexi y o bo h p obabilis ic design p oblems depends
on he p edic ion ho izon in a loga i hmic way, unlike scena io-based app oaches which exhibi
linea dependence. The e icacy o he p oposed app oach is demons a ed wi h a nume ical
example.
Keywo ds: P edic i e con ol, andomized algo i hms, sampling me hods, s ochas ic sys ems,
op imiza ion.
1. INTRODUCTION
Model p edic i e con ol (MPC) is a popula con ol s a -
egy mainly o i s abili y o deal wi h mul i a ia e sys ems
and cons ain s in a sys ema ic ashion. Howe e , he p es-
ence o unce ain ies can signi ican ly deg ade closed-loop
pe o mance, cause iola ion o cons ain s o e en lead
o ins abili ies. These sho comings ha e been add essed
by many esea ch wo ks, since he i s o mula ion o
obus MPC schemes (Campo and Mo a i, 1987) based
on wo s -case analysis. Indeed, adi ional obus MPC
schemes minimize he chosen cos unc ion o he wo s -
case alue o he unce ain ies, which a e assumed o be
de ined in a compac se , and en o ce he cons ain s o
all possible ealiza ions o he unce ain ies. On he o he
Fab izio Dabbene aknowledges he I alian Ins i u e o Technology
and he I alian Minis e o dell’Is uzione, dell’Uni e si `a e della
Rice ca (PRIN 2017 N. 2017S559BB). Teodo o Alamo acknowledges
MEyC Spain (con ac DPI2016-76493-C3-1-R).
1co esponding au ho .
hand, because he wo s -case alue o he unce ain ies
can ha e a e y small p obabili y o occu ence and any
knowledge abou hei p obabili y dis ibu ion is igno ed,
adi ional obus MPC schemes can be e y conse a i e.
An al e na i e o mi iga e such conse a i eness is o o -
mula e s ochas ic MPC p oblems (Mesbah, 2016), which
explici ly conside p obabili y dis ibu ion unc ions, in-
cluding expec a ions o s anda d de ia ions in he cos
unc ions as well as he use o cons ain s ha should
be ul illed in p obabili y, o en called chance cons ain s.
S ochas ic MPC o mula ions ace wo impo an chal-
lenges: (i) he p opaga ion o he s ochas ic unce ain y
h ough he sys em dynamics; and (ii) he conside a ion o
chance cons ain s and ecu si e easibili y. Un o una ely,
exac o mula ions a e in ac able e en in he linea case.
Hence, p e ious wo ks ha e been ocused on di e en sim-
pli ying assump ions, such as conse a i e app oxima ion
o chance cons ain s based on he p opaga ion o he
a iance h ough linea dynamics (Fa ina and Sca olini
(2016), Hewing and Zeilinge (2018)) o ia polynomial
A p obabilis ic alida ion app oach o
penal y unc ion design in
S ochas ic Model P edic i e Con ol
Ma ina Mamma ella ∗1Teodo o Alamo ∗∗ Se gio Lucia ∗∗∗
Fab izio Dabbene ∗
∗Ins i u e o Elec onics, Compu e and Telecommunica ion
Enginee ing, Na ional Resea ch Council o I aly, Tu in, I aly (e-mail:
ma ina.mamma ella@ieii .cn .i , ab izio.dabbene@ ieii .cn .i ).
∗∗ Depa amen o de Ingenie ´ıa de Sis emas y Au om´a ica, Uni e sidad
de Se illa, Escuela Supe io de Ingenie os, Camino de los
Descub imien os s/n, 41092 Se illa, Spain (e-mail: [email protected])
∗∗∗ Eins ein Cen e Digi al Fu u e, Technische Uni e si ¨a Be lin,
Ge many (e-mail: se gio.lucia@ u-be lin.de)
Abs ac : In his pape , we conside a s ochas ic Model P edic i e Con ol able o accoun o
e ec s o addi i e s ochas ic dis u bance wi h unbounded suppo , and equi ing no es ic i e
assump ion on ei he independence no Gaussiani y. We e isi he a he classical app oach
based on penal y unc ions, wi h he aim o designing a con ol scheme ha mee s some
gi en p obabilis ic speci ica ions. The main di e ence wi h p e ious app oaches is ha we do
no ecu o he no ion o p obabilis ic ecu si e easibili y, and hence we do no conside
sepa a ely he un easible case. In pa icula , wo p obabilis ic design p oblems a e en isioned.
The i s andomiza ion p oblem aims o design o line he cons ain se igh ening, ollowing
an app oach inhe i ed om ube-based MPC. Fo he second p obabilis ic scheme, a speci ic
p obabilis ic alida ion app oach is exploi ed o uning he penal y pa ame e , o be selec ed
o line among a ini e- amily o possible alues. The simple algo i hm he e p oposed allows
designing a single con olle , always gua an eeing easibili y o he online op imiza ion p oblem.
The p oposed me hod is shown o be mo e compu a ionally ac able han p e ious schemes.
This is due o he ac ha he sample complexi y o bo h p obabilis ic design p oblems depends
on he p edic ion ho izon in a loga i hmic way, unlike scena io-based app oaches which exhibi
linea dependence. The e icacy o he p oposed app oach is demons a ed wi h a nume ical
example.
Keywo ds: P edic i e con ol, andomized algo i hms, sampling me hods, s ochas ic sys ems,
op imiza ion.
1. INTRODUCTION
Model p edic i e con ol (MPC) is a popula con ol s a -
egy mainly o i s abili y o deal wi h mul i a ia e sys ems
and cons ain s in a sys ema ic ashion. Howe e , he p es-
ence o unce ain ies can signi ican ly deg ade closed-loop
pe o mance, cause iola ion o cons ain s o e en lead
o ins abili ies. These sho comings ha e been add essed
by many esea ch wo ks, since he i s o mula ion o
obus MPC schemes (Campo and Mo a i, 1987) based
on wo s -case analysis. Indeed, adi ional obus MPC
schemes minimize he chosen cos unc ion o he wo s -
case alue o he unce ain ies, which a e assumed o be
de ined in a compac se , and en o ce he cons ain s o
all possible ealiza ions o he unce ain ies. On he o he
Fab izio Dabbene aknowledges he I alian Ins i u e o Technology
and he I alian Minis e o dell’Is uzione, dell’Uni e si `a e della
Rice ca (PRIN 2017 N. 2017S559BB). Teodo o Alamo acknowledges
MEyC Spain (con ac DPI2016-76493-C3-1-R).
1co esponding au ho .
hand, because he wo s -case alue o he unce ain ies
can ha e a e y small p obabili y o occu ence and any
knowledge abou hei p obabili y dis ibu ion is igno ed,
adi ional obus MPC schemes can be e y conse a i e.
An al e na i e o mi iga e such conse a i eness is o o -
mula e s ochas ic MPC p oblems (Mesbah, 2016), which
explici ly conside p obabili y dis ibu ion unc ions, in-
cluding expec a ions o s anda d de ia ions in he cos
unc ions as well as he use o cons ain s ha should
be ul illed in p obabili y, o en called chance cons ain s.
S ochas ic MPC o mula ions ace wo impo an chal-
lenges: (i) he p opaga ion o he s ochas ic unce ain y
h ough he sys em dynamics; and (ii) he conside a ion o
chance cons ain s and ecu si e easibili y. Un o una ely,
exac o mula ions a e in ac able e en in he linea case.
Hence, p e ious wo ks ha e been ocused on di e en sim-
pli ying assump ions, such as conse a i e app oxima ion
o chance cons ain s based on he p opaga ion o he
a iance h ough linea dynamics (Fa ina and Sca olini
(2016), Hewing and Zeilinge (2018)) o ia polynomial
A p obabilis ic alida ion app oach o
penal y unc ion design in
S ochas ic Model P edic i e Con ol
Ma ina Mamma ella ∗1Teodo o Alamo ∗∗ Se gio Lucia ∗∗∗
Fab izio Dabbene ∗
∗Ins i u e o Elec onics, Compu e and Telecommunica ion
Enginee ing, Na ional Resea ch Council o I aly, Tu in, I aly (e-mail:
ma ina.mamma ella@ieii .cn .i , ab izio.dabbene@ ieii .cn .i ).
∗∗ Depa amen o de Ingenie ´ıa de Sis emas y Au om´a ica, Uni e sidad
de Se illa, Escuela Supe io de Ingenie os, Camino de los
Descub imien os s/n, 41092 Se illa, Spain (e-mail: [email protected])
∗∗∗ Eins ein Cen e Digi al Fu u e, Technische Uni e si ¨a Be lin,
Ge many (e-mail: se gio.lucia@ u-be lin.de)
Abs ac : In his pape , we conside a s ochas ic Model P edic i e Con ol able o accoun o
e ec s o addi i e s ochas ic dis u bance wi h unbounded suppo , and equi ing no es ic i e
assump ion on ei he independence no Gaussiani y. We e isi he a he classical app oach
based on penal y unc ions, wi h he aim o designing a con ol scheme ha mee s some
gi en p obabilis ic speci ica ions. The main di e ence wi h p e ious app oaches is ha we do
no ecu o he no ion o p obabilis ic ecu si e easibili y, and hence we do no conside
sepa a ely he un easible case. In pa icula , wo p obabilis ic design p oblems a e en isioned.
The i s andomiza ion p oblem aims o design o line he cons ain se igh ening, ollowing
an app oach inhe i ed om ube-based MPC. Fo he second p obabilis ic scheme, a speci ic
p obabilis ic alida ion app oach is exploi ed o uning he penal y pa ame e , o be selec ed
o line among a ini e- amily o possible alues. The simple algo i hm he e p oposed allows
designing a single con olle , always gua an eeing easibili y o he online op imiza ion p oblem.
The p oposed me hod is shown o be mo e compu a ionally ac able han p e ious schemes.
This is due o he ac ha he sample complexi y o bo h p obabilis ic design p oblems depends
on he p edic ion ho izon in a loga i hmic way, unlike scena io-based app oaches which exhibi
linea dependence. The e icacy o he p oposed app oach is demons a ed wi h a nume ical
example.
Keywo ds: P edic i e con ol, andomized algo i hms, sampling me hods, s ochas ic sys ems,
op imiza ion.
1. INTRODUCTION
Model p edic i e con ol (MPC) is a popula con ol s a -
egy mainly o i s abili y o deal wi h mul i a ia e sys ems
and cons ain s in a sys ema ic ashion. Howe e , he p es-
ence o unce ain ies can signi ican ly deg ade closed-loop
pe o mance, cause iola ion o cons ain s o e en lead
o ins abili ies. These sho comings ha e been add essed
by many esea ch wo ks, since he i s o mula ion o
obus MPC schemes (Campo and Mo a i, 1987) based
on wo s -case analysis. Indeed, adi ional obus MPC
schemes minimize he chosen cos unc ion o he wo s -
case alue o he unce ain ies, which a e assumed o be
de ined in a compac se , and en o ce he cons ain s o
all possible ealiza ions o he unce ain ies. On he o he
Fab izio Dabbene aknowledges he I alian Ins i u e o Technology
and he I alian Minis e o dell’Is uzione, dell’Uni e si `a e della
Rice ca (PRIN 2017 N. 2017S559BB). Teodo o Alamo acknowledges
MEyC Spain (con ac DPI2016-76493-C3-1-R).
1co esponding au ho .
hand, because he wo s -case alue o he unce ain ies
can ha e a e y small p obabili y o occu ence and any
knowledge abou hei p obabili y dis ibu ion is igno ed,
adi ional obus MPC schemes can be e y conse a i e.
An al e na i e o mi iga e such conse a i eness is o o -
mula e s ochas ic MPC p oblems (Mesbah, 2016), which
explici ly conside p obabili y dis ibu ion unc ions, in-
cluding expec a ions o s anda d de ia ions in he cos
unc ions as well as he use o cons ain s ha should
be ul illed in p obabili y, o en called chance cons ain s.
S ochas ic MPC o mula ions ace wo impo an chal-
lenges: (i) he p opaga ion o he s ochas ic unce ain y
h ough he sys em dynamics; and (ii) he conside a ion o
chance cons ain s and ecu si e easibili y. Un o una ely,
exac o mula ions a e in ac able e en in he linea case.
Hence, p e ious wo ks ha e been ocused on di e en sim-
pli ying assump ions, such as conse a i e app oxima ion
o chance cons ain s based on he p opaga ion o he
a iance h ough linea dynamics (Fa ina and Sca olini
(2016), Hewing and Zeilinge (2018)) o ia polynomial
A p obabilis ic alida ion app oach o
penal y unc ion design in
S ochas ic Model P edic i e Con ol
Ma ina Mamma ella ∗1Teodo o Alamo ∗∗ Se gio Lucia ∗∗∗
Fab izio Dabbene ∗
∗Ins i u e o Elec onics, Compu e and Telecommunica ion
Enginee ing, Na ional Resea ch Council o I aly, Tu in, I aly (e-mail:
ma ina.mamma ella@ieii .cn .i , ab izio.dabbene@ ieii .cn .i ).
∗∗ Depa amen o de Ingenie ´ıa de Sis emas y Au om´a ica, Uni e sidad
de Se illa, Escuela Supe io de Ingenie os, Camino de los
Descub imien os s/n, 41092 Se illa, Spain (e-mail: [email protected])
∗∗∗ Eins ein Cen e Digi al Fu u e, Technische Uni e si ¨a Be lin,
Ge many (e-mail: se gio.lucia@ u-be lin.de)
Abs ac : In his pape , we conside a s ochas ic Model P edic i e Con ol able o accoun o
e ec s o addi i e s ochas ic dis u bance wi h unbounded suppo , and equi ing no es ic i e
assump ion on ei he independence no Gaussiani y. We e isi he a he classical app oach
based on penal y unc ions, wi h he aim o designing a con ol scheme ha mee s some
gi en p obabilis ic speci ica ions. The main di e ence wi h p e ious app oaches is ha we do
no ecu o he no ion o p obabilis ic ecu si e easibili y, and hence we do no conside
sepa a ely he un easible case. In pa icula , wo p obabilis ic design p oblems a e en isioned.
The i s andomiza ion p oblem aims o design o line he cons ain se igh ening, ollowing
an app oach inhe i ed om ube-based MPC. Fo he second p obabilis ic scheme, a speci ic
p obabilis ic alida ion app oach is exploi ed o uning he penal y pa ame e , o be selec ed
o line among a ini e- amily o possible alues. The simple algo i hm he e p oposed allows
designing a single con olle , always gua an eeing easibili y o he online op imiza ion p oblem.
The p oposed me hod is shown o be mo e compu a ionally ac able han p e ious schemes.
This is due o he ac ha he sample complexi y o bo h p obabilis ic design p oblems depends
on he p edic ion ho izon in a loga i hmic way, unlike scena io-based app oaches which exhibi
linea dependence. The e icacy o he p oposed app oach is demons a ed wi h a nume ical
example.
Keywo ds: P edic i e con ol, andomized algo i hms, sampling me hods, s ochas ic sys ems,
op imiza ion.
1. INTRODUCTION
Model p edic i e con ol (MPC) is a popula con ol s a -
egy mainly o i s abili y o deal wi h mul i a ia e sys ems
and cons ain s in a sys ema ic ashion. Howe e , he p es-
ence o unce ain ies can signi ican ly deg ade closed-loop
pe o mance, cause iola ion o cons ain s o e en lead
o ins abili ies. These sho comings ha e been add essed
by many esea ch wo ks, since he i s o mula ion o
obus MPC schemes (Campo and Mo a i, 1987) based
on wo s -case analysis. Indeed, adi ional obus MPC
schemes minimize he chosen cos unc ion o he wo s -
case alue o he unce ain ies, which a e assumed o be
de ined in a compac se , and en o ce he cons ain s o
all possible ealiza ions o he unce ain ies. On he o he
Fab izio Dabbene aknowledges he I alian Ins i u e o Technology
and he I alian Minis e o dell’Is uzione, dell’Uni e si `a e della
Rice ca (PRIN 2017 N. 2017S559BB). Teodo o Alamo acknowledges
MEyC Spain (con ac DPI2016-76493-C3-1-R).
1co esponding au ho .
hand, because he wo s -case alue o he unce ain ies
can ha e a e y small p obabili y o occu ence and any
knowledge abou hei p obabili y dis ibu ion is igno ed,
adi ional obus MPC schemes can be e y conse a i e.
An al e na i e o mi iga e such conse a i eness is o o -
mula e s ochas ic MPC p oblems (Mesbah, 2016), which
explici ly conside p obabili y dis ibu ion unc ions, in-
cluding expec a ions o s anda d de ia ions in he cos
unc ions as well as he use o cons ain s ha should
be ul illed in p obabili y, o en called chance cons ain s.
S ochas ic MPC o mula ions ace wo impo an chal-
lenges: (i) he p opaga ion o he s ochas ic unce ain y
h ough he sys em dynamics; and (ii) he conside a ion o
chance cons ain s and ecu si e easibili y. Un o una ely,
exac o mula ions a e in ac able e en in he linea case.
Hence, p e ious wo ks ha e been ocused on di e en sim-
pli ying assump ions, such as conse a i e app oxima ion
o chance cons ain s based on he p opaga ion o he
a iance h ough linea dynamics (Fa ina and Sca olini
(2016), Hewing and Zeilinge (2018)) o ia polynomial
A p obabilis ic alida ion app oach o
penal y unc ion design in
S ochas ic Model P edic i e Con ol
Ma ina Mamma ella ∗1Teodo o Alamo ∗∗ Se gio Lucia ∗∗∗
Fab izio Dabbene ∗
∗Ins i u e o Elec onics, Compu e and Telecommunica ion
Enginee ing, Na ional Resea ch Council o I aly, Tu in, I aly (e-mail:
ma ina.mamma ella@ieii .cn .i , ab izio.dabbene@ ieii .cn .i ).
∗∗ Depa amen o de Ingenie ´ıa de Sis emas y Au om´a ica, Uni e sidad
de Se illa, Escuela Supe io de Ingenie os, Camino de los
Descub imien os s/n, 41092 Se illa, Spain (e-mail: [email protected])
∗∗∗ Eins ein Cen e Digi al Fu u e, Technische Uni e si ¨a Be lin,
Ge many (e-mail: se gio.lucia@ u-be lin.de)
Abs ac : In his pape , we conside a s ochas ic Model P edic i e Con ol able o accoun o
e ec s o addi i e s ochas ic dis u bance wi h unbounded suppo , and equi ing no es ic i e
assump ion on ei he independence no Gaussiani y. We e isi he a he classical app oach
based on penal y unc ions, wi h he aim o designing a con ol scheme ha mee s some
gi en p obabilis ic speci ica ions. The main di e ence wi h p e ious app oaches is ha we do
no ecu o he no ion o p obabilis ic ecu si e easibili y, and hence we do no conside
sepa a ely he un easible case. In pa icula , wo p obabilis ic design p oblems a e en isioned.
The i s andomiza ion p oblem aims o design o line he cons ain se igh ening, ollowing
an app oach inhe i ed om ube-based MPC. Fo he second p obabilis ic scheme, a speci ic
p obabilis ic alida ion app oach is exploi ed o uning he penal y pa ame e , o be selec ed
o line among a ini e- amily o possible alues. The simple algo i hm he e p oposed allows
designing a single con olle , always gua an eeing easibili y o he online op imiza ion p oblem.
The p oposed me hod is shown o be mo e compu a ionally ac able han p e ious schemes.
This is due o he ac ha he sample complexi y o bo h p obabilis ic design p oblems depends
on he p edic ion ho izon in a loga i hmic way, unlike scena io-based app oaches which exhibi
linea dependence. The e icacy o he p oposed app oach is demons a ed wi h a nume ical
example.
Keywo ds: P edic i e con ol, andomized algo i hms, sampling me hods, s ochas ic sys ems,
op imiza ion.
1. INTRODUCTION
Model p edic i e con ol (MPC) is a popula con ol s a -
egy mainly o i s abili y o deal wi h mul i a ia e sys ems
and cons ain s in a sys ema ic ashion. Howe e , he p es-
ence o unce ain ies can signi ican ly deg ade closed-loop
pe o mance, cause iola ion o cons ain s o e en lead
o ins abili ies. These sho comings ha e been add essed
by many esea ch wo ks, since he i s o mula ion o
obus MPC schemes (Campo and Mo a i, 1987) based
on wo s -case analysis. Indeed, adi ional obus MPC
schemes minimize he chosen cos unc ion o he wo s -
case alue o he unce ain ies, which a e assumed o be
de ined in a compac se , and en o ce he cons ain s o
all possible ealiza ions o he unce ain ies. On he o he
Fab izio Dabbene aknowledges he I alian Ins i u e o Technology
and he I alian Minis e o dell’Is uzione, dell’Uni e si `a e della
Rice ca (PRIN 2017 N. 2017S559BB). Teodo o Alamo acknowledges
MEyC Spain (con ac DPI2016-76493-C3-1-R).
1co esponding au ho .
hand, because he wo s -case alue o he unce ain ies
can ha e a e y small p obabili y o occu ence and any
knowledge abou hei p obabili y dis ibu ion is igno ed,
adi ional obus MPC schemes can be e y conse a i e.
An al e na i e o mi iga e such conse a i eness is o o -
mula e s ochas ic MPC p oblems (Mesbah, 2016), which
explici ly conside p obabili y dis ibu ion unc ions, in-
cluding expec a ions o s anda d de ia ions in he cos
unc ions as well as he use o cons ain s ha should
be ul illed in p obabili y, o en called chance cons ain s.
S ochas ic MPC o mula ions ace wo impo an chal-
lenges: (i) he p opaga ion o he s ochas ic unce ain y
h ough he sys em dynamics; and (ii) he conside a ion o
chance cons ain s and ecu si e easibili y. Un o una ely,
exac o mula ions a e in ac able e en in he linea case.
Hence, p e ious wo ks ha e been ocused on di e en sim-
pli ying assump ions, such as conse a i e app oxima ion
o chance cons ain s based on he p opaga ion o he
a iance h ough linea dynamics (Fa ina and Sca olini
(2016), Hewing and Zeilinge (2018)) o ia polynomial
A p obabilis ic alida ion app oach o
penal y unc ion design in
S ochas ic Model P edic i e Con ol
Ma ina Mamma ella ∗1Teodo o Alamo ∗∗ Se gio Lucia ∗∗∗
Fab izio Dabbene ∗
∗Ins i u e o Elec onics, Compu e and Telecommunica ion
Enginee ing, Na ional Resea ch Council o I aly, Tu in, I aly (e-mail:
ma ina.mamma el[email p o ec ed], ab izio.dabbene@ ieii .cn .i ).
∗∗ Depa amen o de Ingenie ´ıa de Sis emas y Au om´a ica, Uni e sidad
de Se illa, Escuela Supe io de Ingenie os, Camino de los
Descub imien os s/n, 41092 Se illa, Spain (e-mail: [email p o ec ed])
∗∗∗ Eins ein Cen e Digi al Fu u e, Technische Uni e si ¨a Be lin,
Ge many (e-mail: se gio.lucia@ u-be lin.de)
Abs ac : In his pape , we conside a s ochas ic Model P edic i e Con ol able o accoun o
e ec s o addi i e s ochas ic dis u bance wi h unbounded suppo , and equi ing no es ic i e
assump ion on ei he independence no Gaussiani y. We e isi he a he classical app oach
based on penal y unc ions, wi h he aim o designing a con ol scheme ha mee s some
gi en p obabilis ic speci ica ions. The main di e ence wi h p e ious app oaches is ha we do
no ecu o he no ion o p obabilis ic ecu si e easibili y, and hence we do no conside
sepa a ely he un easible case. In pa icula , wo p obabilis ic design p oblems a e en isioned.
The i s andomiza ion p oblem aims o design o line he cons ain se igh ening, ollowing
an app oach inhe i ed om ube-based MPC. Fo he second p obabilis ic scheme, a speci ic
p obabilis ic alida ion app oach is exploi ed o uning he penal y pa ame e , o be selec ed
o line among a ini e- amily o possible alues. The simple algo i hm he e p oposed allows
designing a single con olle , always gua an eeing easibili y o he online op imiza ion p oblem.
The p oposed me hod is shown o be mo e compu a ionally ac able han p e ious schemes.
This is due o he ac ha he sample complexi y o bo h p obabilis ic design p oblems depends
on he p edic ion ho izon in a loga i hmic way, unlike scena io-based app oaches which exhibi
linea dependence. The e icacy o he p oposed app oach is demons a ed wi h a nume ical
example.
Keywo ds: P edic i e con ol, andomized algo i hms, sampling me hods, s ochas ic sys ems,
op imiza ion.
1. INTRODUCTION
Model p edic i e con ol (MPC) is a popula con ol s a -
egy mainly o i s abili y o deal wi h mul i a ia e sys ems
and cons ain s in a sys ema ic ashion. Howe e , he p es-
ence o unce ain ies can signi ican ly deg ade closed-loop
pe o mance, cause iola ion o cons ain s o e en lead
o ins abili ies. These sho comings ha e been add essed
by many esea ch wo ks, since he i s o mula ion o
obus MPC schemes (Campo and Mo a i, 1987) based
on wo s -case analysis. Indeed, adi ional obus MPC
schemes minimize he chosen cos unc ion o he wo s -
case alue o he unce ain ies, which a e assumed o be
de ined in a compac se , and en o ce he cons ain s o
all possible ealiza ions o he unce ain ies. On he o he
Fab izio Dabbene aknowledges he I alian Ins i u e o Technology
and he I alian Minis e o dell’Is uzione, dell’Uni e si `a e della
Rice ca (PRIN 2017 N. 2017S559BB). Teodo o Alamo acknowledges
MEyC Spain (con ac DPI2016-76493-C3-1-R).
1co esponding au ho .
hand, because he wo s -case alue o he unce ain ies
can ha e a e y small p obabili y o occu ence and any
knowledge abou hei p obabili y dis ibu ion is igno ed,
adi ional obus MPC schemes can be e y conse a i e.
An al e na i e o mi iga e such conse a i eness is o o -
mula e s ochas ic MPC p oblems (Mesbah, 2016), which
explici ly conside p obabili y dis ibu ion unc ions, in-
cluding expec a ions o s anda d de ia ions in he cos
unc ions as well as he use o cons ain s ha should
be ul illed in p obabili y, o en called chance cons ain s.
S ochas ic MPC o mula ions ace wo impo an chal-
lenges: (i) he p opaga ion o he s ochas ic unce ain y
h ough he sys em dynamics; and (ii) he conside a ion o
chance cons ain s and ecu si e easibili y. Un o una ely,
exac o mula ions a e in ac able e en in he linea case.
Hence, p e ious wo ks ha e been ocused on di e en sim-
pli ying assump ions, such as conse a i e app oxima ion
o chance cons ain s based on he p opaga ion o he
a iance h ough linea dynamics (Fa ina and Sca olini
(2016), Hewing and Zeilinge (2018)) o ia polynomial
A p obabilis ic alida ion app oach o
penal y unc ion design in
S ochas ic Model P edic i e Con ol
Ma ina Mamma ella ∗1Teodo o Alamo ∗∗ Se gio Lucia ∗∗∗
Fab izio Dabbene
∗
∗Ins i u e o Elec onics, Compu e and Telecommunica ion
Enginee ing, Na ional Resea ch Council o I aly, Tu in, I aly (e-mail:
ma ina.mamma ella@ieii .cn .i , ab izio.dabbene@ ieii .cn .i ).
∗∗ Depa amen o de Ingenie ´ıa de Sis emas y Au om´a ica, Uni e sidad
de Se illa, Escuela Supe io de Ingenie os, Camino de los
Descub imien os s/n, 41092 Se illa, Spain (e-mail: [email protected])
∗∗∗ Eins ein Cen e Digi al Fu u e, Technische Uni e si ¨a Be lin,
Ge many (e-mail: se gio.lucia@ u-be lin.de)
Abs ac : In his pape , we conside a s ochas ic Model P edic i e Con ol able o accoun o
e ec s o addi i e s ochas ic dis u bance wi h unbounded suppo , and equi ing no es ic i e
assump ion on ei he independence no Gaussiani y. We e isi he a he classical app oach
based on penal y unc ions, wi h he aim o designing a con ol scheme ha mee s some
gi en p obabilis ic speci ica ions. The main di e ence wi h p e ious app oaches is ha we do
no ecu o he no ion o p obabilis ic ecu si e easibili y, and hence we do no conside
sepa a ely he un easible case. In pa icula , wo p obabilis ic design p oblems a e en isioned.
The i s andomiza ion p oblem aims o design o line he cons ain se igh ening, ollowing
an app oach inhe i ed om ube-based MPC. Fo he second p obabilis ic scheme, a speci ic
p obabilis ic alida ion app oach is exploi ed o uning he penal y pa ame e , o be selec ed
o line among a ini e- amily o possible alues. The simple algo i hm he e p oposed allows
designing a single con olle , always gua an eeing easibili y o he online op imiza ion p oblem.
The p oposed me hod is shown o be mo e compu a ionally ac able han p e ious schemes.
This is due o he ac ha he sample complexi y o bo h p obabilis ic design p oblems depends
on he p edic ion ho izon in a loga i hmic way, unlike scena io-based app oaches which exhibi
linea dependence. The e icacy o he p oposed app oach is demons a ed wi h a nume ical
example.
Keywo ds: P edic i e con ol, andomized algo i hms, sampling me hods, s ochas ic sys ems,
op imiza ion.
1. INTRODUCTION
Model p edic i e con ol (MPC) is a popula con ol s a -
egy mainly o i s abili y o deal wi h mul i a ia e sys ems
and cons ain s in a sys ema ic ashion. Howe e , he p es-
ence o unce ain ies can signi ican ly deg ade closed-loop
pe o mance, cause iola ion o cons ain s o e en lead
o ins abili ies. These sho comings ha e been add essed
by many esea ch wo ks, since he i s o mula ion o
obus MPC schemes (Campo and Mo a i, 1987) based
on wo s -case analysis. Indeed, adi ional obus MPC
schemes minimize he chosen cos unc ion o he wo s -
case alue o he unce ain ies, which a e assumed o be
de ined in a compac se , and en o ce he cons ain s o
all possible ealiza ions o he unce ain ies. On he o he
Fab izio Dabbene aknowledges he I alian Ins i u e o Technology
and he I alian Minis e o dell’Is uzione, dell’Uni e si `a e della
Rice ca (PRIN 2017 N. 2017S559BB). Teodo o Alamo acknowledges
MEyC Spain (con ac DPI2016-76493-C3-1-R).
1co esponding au ho .
hand, because he wo s -case alue o he unce ain ies
can ha e a e y small p obabili y o occu ence and any
knowledge abou hei p obabili y dis ibu ion is igno ed,
adi ional obus MPC schemes can be e y conse a i e.
An al e na i e o mi iga e such conse a i eness is o o -
mula e s ochas ic MPC p oblems (Mesbah, 2016), which
explici ly conside p obabili y dis ibu ion unc ions, in-
cluding expec a ions o s anda d de ia ions in he cos
unc ions as well as he use o cons ain s ha should
be ul illed in p obabili y, o en called chance cons ain s.
S ochas ic MPC o mula ions ace wo impo an chal-
lenges: (i) he p opaga ion o he s ochas ic unce ain y
h ough he sys em dynamics; and (ii) he conside a ion o
chance cons ain s and ecu si e easibili y. Un o una ely,
exac o mula ions a e in ac able e en in he linea case.
Hence, p e ious wo ks ha e been ocused on di e en sim-
pli ying assump ions, such as conse a i e app oxima ion
o chance cons ain s based on he p opaga ion o he
a iance h ough linea dynamics (Fa ina and Sca olini
(2016), Hewing and Zeilinge (2018)) o ia polynomial
A p obabilis ic alida ion app oach o
penal y unc ion design in
S ochas ic Model P edic i e Con ol
Ma ina Mamma ella ∗1Teodo o Alamo ∗∗ Se gio Lucia ∗∗∗
Fab izio Dabbene ∗
∗Ins i u e o Elec onics, Compu e and Telecommunica ion
Enginee ing, Na ional Resea ch Council o I aly, Tu in, I aly (e-mail:
ma ina.mamma ella@ieii .cn .i , ab izio.dabbene@ ieii .cn .i ).
∗∗ Depa amen o de Ingenie ´ıa de Sis emas y Au om´a ica, Uni e sidad
de Se illa, Escuela Supe io de Ingenie os, Camino de los
Descub imien os s/n, 41092 Se illa, Spain (e-mail: [email protected])
∗∗∗ Eins ein Cen e Digi al Fu u e, Technische Uni e si ¨a Be lin,
Ge many (e-mail: se gio.lucia@ u-be lin.de)
Abs ac : In his pape , we conside a s ochas ic Model P edic i e Con ol able o accoun o
e ec s o addi i e s ochas ic dis u bance wi h unbounded suppo , and equi ing no es ic i e
assump ion on ei he independence no Gaussiani y. We e isi he a he classical app oach
based on penal y unc ions, wi h he aim o designing a con ol scheme ha mee s some
gi en p obabilis ic speci ica ions. The main di e ence wi h p e ious app oaches is ha we do
no ecu o he no ion o p obabilis ic ecu si e easibili y, and hence we do no conside
sepa a ely he un easible case. In pa icula , wo p obabilis ic design p oblems a e en isioned.
The i s andomiza ion p oblem aims o design o line he cons ain se igh ening, ollowing
an app oach inhe i ed om ube-based MPC. Fo he second p obabilis ic scheme, a speci ic
p obabilis ic alida ion app oach is exploi ed o uning he penal y pa ame e , o be selec ed
o line among a ini e- amily o possible alues. The simple algo i hm he e p oposed allows
designing a single con olle , always gua an eeing easibili y o he online op imiza ion p oblem.
The p oposed me hod is shown o be mo e compu a ionally ac able han p e ious schemes.
This is due o he ac ha he sample complexi y o bo h p obabilis ic design p oblems depends
on he p edic ion ho izon in a loga i hmic way, unlike scena io-based app oaches which exhibi
linea dependence. The e icacy o he p oposed app oach is demons a ed wi h a nume ical
example.
Keywo ds: P edic i e con ol, andomized algo i hms, sampling me hods, s ochas ic sys ems,
op imiza ion.
1. INTRODUCTION
Model p edic i e con ol (MPC) is a popula con ol s a -
egy mainly o i s abili y o deal wi h mul i a ia e sys ems
and cons ain s in a sys ema ic ashion. Howe e , he p es-
ence o unce ain ies can signi ican ly deg ade closed-loop
pe o mance, cause iola ion o cons ain s o e en lead
o ins abili ies. These sho comings ha e been add essed
by many esea ch wo ks, since he i s o mula ion o
obus MPC schemes (Campo and Mo a i, 1987) based
on wo s -case analysis. Indeed, adi ional obus MPC
schemes minimize he chosen cos unc ion o he wo s -
case alue o he unce ain ies, which a e assumed o be
de ined in a compac se , and en o ce he cons ain s o
all possible ealiza ions o he unce ain ies. On he o he
Fab izio Dabbene aknowledges he I alian Ins i u e o Technology
and he I alian Minis e o dell’Is uzione, dell’Uni e si `a e della
Rice ca (PRIN 2017 N. 2017S559BB). Teodo o Alamo acknowledges
MEyC Spain (con ac DPI2016-76493-C3-1-R).
1co esponding au ho .
hand, because he wo s -case alue o he unce ain ies
can ha e a e y small p obabili y o occu ence and any
knowledge abou hei p obabili y dis ibu ion is igno ed,
adi ional obus MPC schemes can be e y conse a i e.
An al e na i e o mi iga e such conse a i eness is o o -
mula e s ochas ic MPC p oblems (Mesbah, 2016), which
explici ly conside p obabili y dis ibu ion unc ions, in-
cluding expec a ions o s anda d de ia ions in he cos
unc ions as well as he use o cons ain s ha should
be ul illed in p obabili y, o en called chance cons ain s.
S ochas ic MPC o mula ions ace wo impo an chal-
lenges: (i) he p opaga ion o he s ochas ic unce ain y
h ough he sys em dynamics; and (ii) he conside a ion o
chance cons ain s and ecu si e easibili y. Un o una ely,
exac o mula ions a e in ac able e en in he linea case.
Hence, p e ious wo ks ha e been ocused on di e en sim-
pli ying assump ions, such as conse a i e app oxima ion
o chance cons ain s based on he p opaga ion o he
a iance h ough linea dynamics (Fa ina and Sca olini
(2016), Hewing and Zeilinge (2018)) o ia polynomial
11272 Ma ina Mamma ella e al. / IFAC Pape sOnLine 53-2 (2020) 11271–11276
chaos expansions (Mesbah, 2016). O he o mula ions use
he scena io app oach (P andini e al. (2012), Cala io e
and Fagiano (2013), (Schildbach e al., 2014), Ma gellos
e al. (2014)), and also an o line scena io se ing ha leads
o a educed numbe o samples and lowe compu a ional
load as shown in Lo enzen e al. (2017), Mamma ella
e al. (2018). Ano he p oblem o deal wi h in a s ochas ic
MPC se ing is ela ed o gua an eeing ecu si e easibil-
i y. To o e come his di icul y, he app oach p esen ed
in Lo enzen e al. (2017) employs cons ain igh ening
o gua an ee obus cons ain sa is ac ion o bounded
unce ain y whe eas Fleming and Cannon (2019) uses i s -
s ep chance cons ain and applies obus cons ain s o
he es o he ho izon.
The main con ibu ion o his wo k is he achie emen
o he desi ed closed-loop gua an ees by means o p oba-
bilis ic alida ion echniques (Tempo e al., 1997), (Alamo
e al., 2015), which a e used a wo di e en le els. The
i s use o p obabilis ic alida ion is o compu e o line a
cons ain igh ening, ollowing he s ochas ic ube-based
MPC app oach p oposed in (Lo enzen e al., 2016), bu us-
ing p obabilis ic alida ion se ing ins ead o he scena io
app oach. Secondly, o gua an ee ecu si e easibili y, we
elax he cons ain s using a penal y unc ion me hod
Ke igan and Maciejowski (2000) and, ollowing ideas p e-
sen ed in Ka g e al. (2019), we pe o m an o line p ob-
abilis ic design o he penal y pa ame e , selec ed among
a ini e- amily o alues, so ha he desi ed p obabilis ic
gua an ees o he closed-loop cons ain sa is ac ion a e
ul illed.
The p oposed app oach leads o an MPC o mula ion ha
is always easible and i can ob ain a e i iable closed-
loop pe o mance. An impo an me i o ou me hod
is ha no assump ions on independence o Gaussiani y
o he s ochas ic a iables a e necessa y. In addi ion,
he ob ained sample complexi y does no depend on he
design space as in he scena io app oach (Cala io e and
Fagiano, 2013), o on quan i ies di icul o compu e in
gene al such as he Vapnik–Che onenkis (VC) dimension
(Lo enzen e al., 2017). As a esul , he sample complexi y
depends on he p edic ion ho izon only in a loga i hmic
way, signi ican ly educing he numbe o samples o d aw
and consequen ly he compu a ional load. The po en ial
o he app oach has been shown ia a nume ical example
in Mamma ella e al. (2020).
The emainde o he pape is o ganized as ollows. Sec-
ion 2 desc ibes he ma hema ical p oblem se up. Sec ion 3
bounds he e ec o he dis u bances while Sec ion 4
desc ibes how o design a p ope igh ening o he con-
s ain s. Sec ion 5 desc ibes he penal y unc ion me hod
used o ob ain always a easible op imiza ion p oblem and
Sec ion 6 discusses he choice o he penal y pa ame e .
Main conclusions o he wo k a e p esen ed in Sec ion 7.
No a ion: The se N>0deno es he posi i e in ege s, he
se N≥0={0}∪N>0 he non-nega i e in ege s, and Nb
a he
in ege s in e al [a, b]. Simila ly R>0(R≥0) o posi i e eal
numbe s. We use xk o he (measu ed) s a e a ime kand
x|k o he s a e p edic ed s eps ahead a ime k. Posi i e
(semi)de ini e ma ices Aa e deno ed A0(A0) and
x2
A
.
=xTAx. Fo ec o s, x0(x0) is in ended
componen -wise. Callig aphic uppe -case le e s, e.g. A,
deno e se s. PAdeno es he p obabilis ic dis ibu ion o
a andom a iable a∈A. Sequence o scala s/ ec o s a e
deno ed wi h bold lowe -case le e s, i.e. . Gi en a ec o
α=[α1,...,α
nα]T∈Rnα, hen α+is a scala de ined as
α+.
=
nα

i=1
max{0,α
i}.
2. PROBLEM SETUP
Le us conside he ollowing linea ime-in a ian sys em
a ec ed by pe sis en , addi i e dis u bance ζk∈Rnx
xk+1 =Axk+Buk+ζk,∀k∈N≥0(1)
whe e xk∈Rnxis he s a e a iable a ime k,uk∈Rnuis
he con ol inpu , and Aand Ba e ma ices o app op ia e
dimensions. No assump ion on nei he independence no
Gaussiani y a e made on he s ochas ic dis u bance ζk.
Mo eo e , bo h s a e and inpu a e cons ained in com-
pac se s Xand U, espec i ely, and he co esponding
cons ain s can be de ined in a compac o m as
Cxk+Dukh. (2)
The con ol objec i e is o design a s abilizing eceding
ho izon con ol, which gua an ees cons ain sa is ac ion
in a p obabilis ic se ing. We will conside he ollowing
quad a ic s age cos
L(xk,u
k).
=xk2
Q+uk2
R,(3)
whe e Q∈Rnx×nx,Q0, R∈Rnu×nu,R0. To
sol e he con ol p oblem, a s ochas ic MPC algo i hm
is conside ed whe e, as ypical o p edic i e schemes, he
op imal con ol p oblem is sol ed epea edly o e a ini e
ho izon N, bu only he i s con ol ac ion o he op imal
sequence is implemen ed (see Mayne e al. (2000) o a
me iculous e iew on MPC). The p oposed con olle is
designed by means o a wo-s ep p ocedu e:
(i) Using a sampling me hod, we i s bound he e ec
o dis u bances in a p obabilis ic manne . This allows
us o o mula e a nominal model p edic i e con olle
ha add esses he chance cons ain s issue (Sec ion 3
and Sec ion 4).
(ii) To a oid in easibili y o he p oposed MPC app oach,
we ew i e he con olle using a penal y cos scheme.
In pa icula , he penal y ac o is adjus ed using
sampling in such a way ha he esul ing con olle
mee s online he p obabilis ic speci ica ions on he
cons ain s sa is ac ion (Sec ion 5 and Sec ion 6).
3. PROBABILISTIC UPPER BOUNDS OF THE
EFFECT OF DISTURBANCES
As i is common in obus and s ochas ic MPC, le us
conside he s a e o he sys em x|k, p edic ed s eps
ahead om ime k, spli in o a de e minis ic, nominal pa
z|kand an e o pa e|kas
x|k=z|k+e|k.(4)
Then, a pa ame ized eedback policy o he o m
u|k= |k+Kx|k,∀∈NN−1
0,(5)
is conside ed whe e he eedback gain ma ix Kis quad a -
ically s abilizing o he sys em (1).
Ma ina Mamma ella e al. / IFAC Pape sOnLine 53-2 (2020) 11271–11276 11273
chaos expansions (Mesbah, 2016). O he o mula ions use
he scena io app oach (P andini e al. (2012), Cala io e
and Fagiano (2013), (Schildbach e al., 2014), Ma gellos
e al. (2014)), and also an o line scena io se ing ha leads
o a educed numbe o samples and lowe compu a ional
load as shown in Lo enzen e al. (2017), Mamma ella
e al. (2018). Ano he p oblem o deal wi h in a s ochas ic
MPC se ing is ela ed o gua an eeing ecu si e easibil-
i y. To o e come his di icul y, he app oach p esen ed
in Lo enzen e al. (2017) employs cons ain igh ening
o gua an ee obus cons ain sa is ac ion o bounded
unce ain y whe eas Fleming and Cannon (2019) uses i s -
s ep chance cons ain and applies obus cons ain s o
he es o he ho izon.
The main con ibu ion o his wo k is he achie emen
o he desi ed closed-loop gua an ees by means o p oba-
bilis ic alida ion echniques (Tempo e al., 1997), (Alamo
e al., 2015), which a e used a wo di e en le els. The
i s use o p obabilis ic alida ion is o compu e o line a
cons ain igh ening, ollowing he s ochas ic ube-based
MPC app oach p oposed in (Lo enzen e al., 2016), bu us-
ing p obabilis ic alida ion se ing ins ead o he scena io
app oach. Secondly, o gua an ee ecu si e easibili y, we
elax he cons ain s using a penal y unc ion me hod
Ke igan and Maciejowski (2000) and, ollowing ideas p e-
sen ed in Ka g e al. (2019), we pe o m an o line p ob-
abilis ic design o he penal y pa ame e , selec ed among
a ini e- amily o alues, so ha he desi ed p obabilis ic
gua an ees o he closed-loop cons ain sa is ac ion a e
ul illed.
The p oposed app oach leads o an MPC o mula ion ha
is always easible and i can ob ain a e i iable closed-
loop pe o mance. An impo an me i o ou me hod
is ha no assump ions on independence o Gaussiani y
o he s ochas ic a iables a e necessa y. In addi ion,
he ob ained sample complexi y does no depend on he
design space as in he scena io app oach (Cala io e and
Fagiano, 2013), o on quan i ies di icul o compu e in
gene al such as he Vapnik–Che onenkis (VC) dimension
(Lo enzen e al., 2017). As a esul , he sample complexi y
depends on he p edic ion ho izon only in a loga i hmic
way, signi ican ly educing he numbe o samples o d aw
and consequen ly he compu a ional load. The po en ial
o he app oach has been shown ia a nume ical example
in Mamma ella e al. (2020).
The emainde o he pape is o ganized as ollows. Sec-
ion 2 desc ibes he ma hema ical p oblem se up. Sec ion 3
bounds he e ec o he dis u bances while Sec ion 4
desc ibes how o design a p ope igh ening o he con-
s ain s. Sec ion 5 desc ibes he penal y unc ion me hod
used o ob ain always a easible op imiza ion p oblem and
Sec ion 6 discusses he choice o he penal y pa ame e .
Main conclusions o he wo k a e p esen ed in Sec ion 7.
No a ion: The se N>0deno es he posi i e in ege s, he
se N≥0={0}∪N>0 he non-nega i e in ege s, and Nb
a he
in ege s in e al [a, b]. Simila ly R>0(R≥0) o posi i e eal
numbe s. We use xk o he (measu ed) s a e a ime kand
x|k o he s a e p edic ed s eps ahead a ime k. Posi i e
(semi)de ini e ma ices Aa e deno ed A0(A0) and
x2
A
.
=xTAx. Fo ec o s, x0(x0) is in ended
componen -wise. Callig aphic uppe -case le e s, e.g. A,
deno e se s. PAdeno es he p obabilis ic dis ibu ion o
a andom a iable a∈A. Sequence o scala s/ ec o s a e
deno ed wi h bold lowe -case le e s, i.e. . Gi en a ec o
α=[α1,...,α
nα]T∈Rnα, hen α+is a scala de ined as
α+.
=
nα

i=1
max{0,α
i}.
2. PROBLEM SETUP
Le us conside he ollowing linea ime-in a ian sys em
a ec ed by pe sis en , addi i e dis u bance ζk∈Rnx
xk+1 =Axk+Buk+ζk,∀k∈N≥0(1)
whe e xk∈Rnxis he s a e a iable a ime k,uk∈Rnuis
he con ol inpu , and Aand Ba e ma ices o app op ia e
dimensions. No assump ion on nei he independence no
Gaussiani y a e made on he s ochas ic dis u bance ζk.
Mo eo e , bo h s a e and inpu a e cons ained in com-
pac se s Xand U, espec i ely, and he co esponding
cons ain s can be de ined in a compac o m as
Cxk+Dukh. (2)
The con ol objec i e is o design a s abilizing eceding
ho izon con ol, which gua an ees cons ain sa is ac ion
in a p obabilis ic se ing. We will conside he ollowing
quad a ic s age cos
L(xk,u
k).
=xk2
Q+uk2
R,(3)
whe e Q∈Rnx×nx,Q0, R∈Rnu×nu,R0. To
sol e he con ol p oblem, a s ochas ic MPC algo i hm
is conside ed whe e, as ypical o p edic i e schemes, he
op imal con ol p oblem is sol ed epea edly o e a ini e
ho izon N, bu only he i s con ol ac ion o he op imal
sequence is implemen ed (see Mayne e al. (2000) o a
me iculous e iew on MPC). The p oposed con olle is
designed by means o a wo-s ep p ocedu e:
(i) Using a sampling me hod, we i s bound he e ec
o dis u bances in a p obabilis ic manne . This allows
us o o mula e a nominal model p edic i e con olle
ha add esses he chance cons ain s issue (Sec ion 3
and Sec ion 4).
(ii) To a oid in easibili y o he p oposed MPC app oach,
we ew i e he con olle using a penal y cos scheme.
In pa icula , he penal y ac o is adjus ed using
sampling in such a way ha he esul ing con olle
mee s online he p obabilis ic speci ica ions on he
cons ain s sa is ac ion (Sec ion 5 and Sec ion 6).
3. PROBABILISTIC UPPER BOUNDS OF THE
EFFECT OF DISTURBANCES
As i is common in obus and s ochas ic MPC, le us
conside he s a e o he sys em x|k, p edic ed s eps
ahead om ime k, spli in o a de e minis ic, nominal pa
z|kand an e o pa e|kas
x|k=z|k+e|k.(4)
Then, a pa ame ized eedback policy o he o m
u|k= |k+Kx|k,∀∈NN−1
0,(5)
is conside ed whe e he eedback gain ma ix Kis quad a -
ically s abilizing o he sys em (1).
Hence, conside ing he eedback policy (5), he sys em
dynamics in (1) along he p edic ion ho izon Ncan be
ew i en in e ms o nominal and e o dynamics as
z+1|k=AKz|k+B |k,z
0|k=xk,(6a)
e+1|k=AKe|k+ζ|k,e
0|k=0,(6b)
whe e ζ|k
.
=ζ+kand AK
.
=A+BK. Now, conside ing
he s a e decomposi ion in (4) and he eedback policy (5),
he cons ain (2) can be ew i en as
CKz|k+D |k+CKe|kh, ∀∈IN N−1
=0 (7)
whe e CK
.
=(C+DK). In absence o dis u bances, e|k=
0, o all ∈IN N−1
=0 and, consequen ly, he cons ain s
gi en in (7) would be equi alen o
CKz|k+D |kh, ∀∈IN N−1
0.(8)
On he o he hand, in he p esence o andom dis u -
bances, one has o deal wi h he unce ain andom ec o s
CKe|k, wi h ∈IN N−1
=0 ha appea in (7). One possibili y
is o p obabilis ically uppe bound hose e ms. This is
p ecisely he objec i e o he emaining o his sec ion.
3.1 P elimina ies: P obabilis ic uppe bound o a andom
a iable
We i s p esen a gene aliza ion o he no ion o he
maximum o a collec ion o scala s, bo owed om he ield
o o de s a is ics (Ahsanullah e al., 2013; A nold e al.,
1992). This will allow us o educe he conse a i eness
ha ollows om he use o he s anda d no ion o max
unc ion. See also Sec ion 3 o Alamo e al. (2018).
De ini ion 1. (O de ed Sequence). Gi en a sequence o S
scala s
={ (1), (2),..., (S)}={ (i)}S
i=1,
we deno e wi h +={ (i)
+}S
i=1 he o de ed sequence
ob ained by ea anging he elemen s o in a non-
inc easing o de . Tha is
(1)
+≥ (2)
+≥...≥ (S−1)
+≥ (S)
+.
De ini ion 2. (Gene alized max unc ion · ). Gi en a se-
quence ={ (i)}S
i=1 o Sscala s and he in ege ∈[1,S],
we de ine he gene alized max unc ion   as
 
.
= ( )
+.
whe e { (i)
+}S
i=1 is gi en in De ini ion 1.
Clea ly, applying De ini ion 1, we ha e
 1= (1)
+= max
1≤i≤S (i), S= (S)
+= min
1≤i≤S (i).
Fu he mo e,  2deno es he second la ges alue in ,
 3 he hi d la ges one, e c. We no ice ha he no a ion
  does no need o make explici S, he numbe o
componen s o . The ollowing p ope y, which has been
al eady p o ed in (Alamo e al., 2018, P ope y 3), s a es
ha he gene alized no ion o max unc ion can be used
o p o ide a p obabilis ic uppe bound o a gi en andom
a iable.
P ope y 1. Conside a andom scala a iable ∈Vwi h
p obabilis ic dis ibu ion PV. Suppose ha ={ (i)}S
i=1
is a sequence o Sindependen iden ically dis ibu ed
(i.i.d.) scala s ha ha e been d awn acco ding o PV.
Then, wi h p obabili y no smalle han 1 −δ,
PV{ >  }≤,
p o ided ha ∈IN S
1and
−1

m=0 S
mm(1 −)S−m≤δ. (9)
Mo eo e , (9) is sa is ied i
S≥1
 −1 + ln 1
δ+2( −1) ln 1
δ.
P ope y 1 has been al eady used in he con ex o p oba-
bilis ic scaling and alida ion (see Alamo e al. (2019) and
Ka g e al. (2019)). See also Tempo e al. (1997) o he
pa icula iza ion o he esul o he case = 1 and a single
cons ain . In he ollowing sec ion, we will gene alize his
esul in such a way ha i will allow us o add ess he
p obabilis ic igh ening o he con ol cons ain s in o de
o cope wi h he unce ain dis u bances.
3.2 Sample-based p obabilis ic uppe bounds o he e ec
o dis u bances
F om he unce ain dynamics gi en in (6b), we ha e ha
he sequence {e|k}N−1
=0 is comple ely de e mined by he
sequence
ζ={ζ0|k,ζ
1|k,...,ζ
N−1|k}={ζk,ζ
k+1,...,ζ
k+N−1}.
We assume ha ζis a s a iona y andom ec o wi h
p obabili y dis ibu ion PDi n IR nx×N. Since we assume
ha he p obabili y dis ibu ion o ζ∈Dis independen
o sample ime kdue o he s a iona y na u e o andom
sequence ζ, we ha e ha he p obabili y dis ibu ion o
e|kis equal o he p obabili y dis ibu ion o e|0. In o de
o make explici he dependence o e|kon he unce ain
dis u bances, we deno e by {e|k(ζ)}N−1
=0 he sequence o
e o dynamics (6b) co esponding o he sequence ζ.
Now, we gene alize he esul s o P ope y 1 o ob ain
sample-based p obabilis ic uppe bounds o
CK,j e|k(ζ),j∈Nnh
1,∈NN−1
0,
whe e CK,j deno es he j- h ow o ma ix CK.
Theo em 1. Gi en a disca ding pa ame e q, and he
p obabilis ic le els q∈(0,1) and δq∈(0,1), suppose ha
Sqi.i.d. samples {ζ(1),...,ζ(Sq)}a e d awn acco ding o
PD. Le us assume also ha he ec o s
{q0,q
1,...,q
N−1}∈ IR nh×N
a e compu ed using he ollowing exp ession
q,j ={CK,j e|0(ζ(i))}Sq
i=1 q
,∀∈IN N−1
0,∀j∈IN nh
1,
(10)
whe e q,j is he j- h componen o q∈IR nh. Then, wi h
p obabili y no smalle han 1 −δq, we ha e
PD{CK,j e|k(ζ)>q
,j}≤q,∀∈IN N−1
0,∀j∈IN nh
1,
p o ided ha ∈NSq
1and
q−1

m=0 Sq
mm
q(1 −q)Sq−m≤δq
nhN.(11)
11274 Ma ina Mamma ella e al. / IFAC Pape sOnLine 53-2 (2020) 11271–11276
In addi ion, (11) is sa is ied i
Sq≥1
q q−1 + ln nhN
δq
+2( q−1) ln nhN
δq.(12)
P oo o Theo em 1 can be ound in Appendix A o
Mamma ella e al. (2020).
4. SAMPLE-BASED CONSTRAINT TIGHTENING
The objec i e o his sec ion is o p esen a model p edic-
i e con olle wi h a cons ain igh ening based on he
p obabilis ic uppe bounds {q}N−1
=0 ha can be ob ained
om Theo em 1.
Fo a classical MPC scheme wi h semi- eedback s uc u e
u|k=Kx|k+ |k, he ini e ho izon cos J(xk, k) o be
minimized a ime kis de ined as
J(xk, k)=
N−1

=0
(x|k2
Q+Kx|k+ |k2
R)+xN|k2
˜
P,
(13)
whe e xk=[x0|k,...,x
N|k]T, k=[ 0|k,...,
N−1|k]T,
and ˜
Pis he solu ion o he disc e e- ime Ricca i equa ion
Q+KTRK +AT
K˜
PAK=˜
P. (14)
Then, a nominal ini e ho izon op imiza ion p oblem
Pnom(xk) can be de ined as
min
xk, k
J(xk, k) (15a)
s. . x0|k=xk,(15b)
x+1|k=AKx|k+B |k∀∈NN−2
0,(15c)
xN−1|k=AKxN−1|k+B N−1|k,(15d)
CKx|k+D |kh, ∀∈NN−1
0.(15e)
We no ice ha cons ain (15d) is a e minal cons ain
ha o ces xN−1 o be an equilib ium poin o he
nominal sys em. This co esponds o a acking MPC
app oach a ge ing he o igin and en o cing an equilib ium
poin as e minal cons ain . This app oach has he e ec
o enla ging he domain o a ac ion o he o igin (Limon
e al. (2008)).
In his wo k, we inhe i he ypical app oach exploi ed in
ube-based MPC schemes (see e.g. Mayne and Rawlings
(2009)), whe e he con ol objec i e becomes con olling
he nominal dynamics z|kin (6a) by sol ing he ollowing
op imiza ion p oblem subjec o a igh ened e sion o he
nominal cons ain s gi en in (8).
De ini ion 3. (Fini e Ho izon Op imiza ion P oblem wi h
Tigh ened Cons ain s) Gi en an ini ial condi ion z0|k∈
Rnx, wi h z0|k=xk, we o mula e he op imiza ion
p oblem Pq(xk) wi h igh ened cons ain s as
min
zk, k
J(zk, k) (16a)
s. . z0|k=xk,(16b)
z+1|k=AKz|k+B |k,∀∈NN−2
0,(16c)
zN−1|k=AKzN−1|k+B N−1|k,(16d)
CKz|k+D |kh−q,∀∈NN−1
0,(16e)
We deno e wi h XN he se o ini ial condi ions xk
o which p oblem Pq(xk) is easible. Fo e e y xk∈
XNwe deno e he minimize o Pq(xk)by(z∗
k, ∗
k)=
(z∗
0|k,...,z∗
N|k, ∗
0|k,..., ∗
N−1|k).
I is impo an o highligh ha , as in (13), a weigh ed
e minal cos is included o ensu e ha he op imal
cos J(zk, k) is a Lyapuno unc ion o he sys em,
gua an eeing s abili y. By p ope ly weigh ing he e minal
cos , he domain o a ac ion o his con olle can be
enla ged. See e.g. Limon e al. (2006) and e e ences
he ein.
F om (7) we ha e ha he con ol cons ain s
Cx|k+Du|kh, ∀∈IN N−1
0,
can be ew i en as
CKz|k+Du|kh−CKe|k.(17)
Thus, he igh ened cons ain s (16e) gua an ee he sa -
is ac ion o (17) p o ided ha Cke|kq. We conclude
ha
CKz|k+D |kh−q
Cke|kq⇒Cx|k+Du|kh.
F om he e we in e ha , o e e y easible solu ion (zk, k)
o Pq(xk),
PD{Cx|k+Du|kh}≥PD{CKe|kq}.
Deno ing Cj,Dj, and CK,j he j- h ows o ma ices C,
Dand CK espec i ely, and hj,q,j he j- h componen s
o hand qwe also ob ain
PD{Cjx|k+Dju|k≤hj}≥PD{CK,j e|k≤q,j}.
Gi en δq∈(0,1) and q∈(0,1), Theo em 1 p o ides a way
o ob ain {q}N−1
=0 such ha , wi h p obabili y no smalle
han 1 −δq,
PD{CK,j e|k(ζ)>q
,j}≤q,∀∈IN N−1
0,∀j∈IN nh
1.
Thus, we conclude ha , gi en δq∈(0,1) and q∈(0,1), i
{q}N−1
=0 a e ob ained acco ding o Theo em 1 hen, wi h
p obabili y no smalle han 1 −δq,
PD{Cjx|k+Dju|k≤hj}≥1−q,∀∈IN N−1
0,∀j∈IN nh
1.
We no ice ha he p e ious p obabilis ic bound does no
e e o he closed-loop beha iou , bu o he p edic ion
scheme o he s ochas ic MPC o mula ion. In he ollow-
ing sec ions, we p esen how o design a so cons ained
e sion o he con olle p oposed in (16) in o de o ob ain
closed-loop p obabilis ic gua an ees.
5. SOFT-CONSTRAINED CONTROLLER
The easibili y egion XNo p oblem Pq(·) is o en a
bounded egion a ound he o igin. Mo eo e , i he p ob-
abili y dis ibu ion o he dis u bances has no a ini e
suppo , hen he on-line ecu si e easibili y o Pq(·) can
only be gua an eed in a p obabilis ic way (see, o exam-
ple, Fleming and Cannon (2019)). In o de o ci cum en
his p oblem, we p opose a so cons ained o mula ion
o he op imiza ion p oblem ha de ines he s ochas ic
MPC con olle . The p oposed scheme elies on he no ion
o penal y unc ion (Ke igan and Maciejowski (2000)).
Gi en a penal y ac o ρ>0, and an ini ial condi ion xk,
we de ine he new op imiza ion p oblem Pρ(xk) as
min
zk, k
J(zk, k)+ρ
N−1

=0
CKz|k+D |k−h+q+
(18a)
s. . (16b),(16c),(16d),(18b)
u0|k= 0|k+Kx0|k∈U.(18c)
Ma ina Mamma ella e al. / IFAC Pape sOnLine 53-2 (2020) 11271–11276 11275
In addi ion, (11) is sa is ied i
Sq≥1
q q−1 + ln nhN
δq
+2( q−1) ln nhN
δq.(12)
P oo o Theo em 1 can be ound in Appendix A o
Mamma ella e al. (2020).
4. SAMPLE-BASED CONSTRAINT TIGHTENING
The objec i e o his sec ion is o p esen a model p edic-
i e con olle wi h a cons ain igh ening based on he
p obabilis ic uppe bounds {q}N−1
=0 ha can be ob ained
om Theo em 1.
Fo a classical MPC scheme wi h semi- eedback s uc u e
u|k=Kx|k+ |k, he ini e ho izon cos J(xk, k) o be
minimized a ime kis de ined as
J(xk, k)=
N−1

=0
(x|k2
Q+Kx|k+ |k2
R)+xN|k2
˜
P,
(13)
whe e xk=[x0|k,...,x
N|k]T, k=[ 0|k,...,
N−1|k]T,
and ˜
Pis he solu ion o he disc e e- ime Ricca i equa ion
Q+KTRK +AT
K˜
PAK=˜
P. (14)
Then, a nominal ini e ho izon op imiza ion p oblem
Pnom(xk) can be de ined as
min
xk, k
J(xk, k) (15a)
s. . x0|k=xk,(15b)
x+1|k=AKx|k+B |k∀∈NN−2
0,(15c)
xN−1|k=AKxN−1|k+B N−1|k,(15d)
CKx|k+D |kh, ∀∈NN−1
0.(15e)
We no ice ha cons ain (15d) is a e minal cons ain
ha o ces xN−1 o be an equilib ium poin o he
nominal sys em. This co esponds o a acking MPC
app oach a ge ing he o igin and en o cing an equilib ium
poin as e minal cons ain . This app oach has he e ec
o enla ging he domain o a ac ion o he o igin (Limon
e al. (2008)).
In his wo k, we inhe i he ypical app oach exploi ed in
ube-based MPC schemes (see e.g. Mayne and Rawlings
(2009)), whe e he con ol objec i e becomes con olling
he nominal dynamics z|kin (6a) by sol ing he ollowing
op imiza ion p oblem subjec o a igh ened e sion o he
nominal cons ain s gi en in (8).
De ini ion 3. (Fini e Ho izon Op imiza ion P oblem wi h
Tigh ened Cons ain s) Gi en an ini ial condi ion z0|k∈
Rnx, wi h z0|k=xk, we o mula e he op imiza ion
p oblem Pq(xk) wi h igh ened cons ain s as
min
zk, k
J(zk, k) (16a)
s. . z0|k=xk,(16b)
z+1|k=AKz|k+B |k,∀∈NN−2
0,(16c)
zN−1|k=AKzN−1|k+B N−1|k,(16d)
CKz|k+D |kh−q,∀∈NN−1
0,(16e)
We deno e wi h XN he se o ini ial condi ions xk
o which p oblem Pq(xk) is easible. Fo e e y xk∈
XNwe deno e he minimize o Pq(xk)by(z∗
k, ∗
k)=
(z∗
0|k,...,z∗
N|k, ∗
0|k,..., ∗
N−1|k).
I is impo an o highligh ha , as in (13), a weigh ed
e minal cos is included o ensu e ha he op imal
cos J(zk, k) is a Lyapuno unc ion o he sys em,
gua an eeing s abili y. By p ope ly weigh ing he e minal
cos , he domain o a ac ion o his con olle can be
enla ged. See e.g. Limon e al. (2006) and e e ences
he ein.
F om (7) we ha e ha he con ol cons ain s
Cx|k+Du|kh, ∀∈IN N−1
0,
can be ew i en as
CKz|k+Du|kh−CKe|k.(17)
Thus, he igh ened cons ain s (16e) gua an ee he sa -
is ac ion o (17) p o ided ha Cke|kq. We conclude
ha
CKz|k+D |kh−q
Cke|kq⇒Cx|k+Du|kh.
F om he e we in e ha , o e e y easible solu ion (zk, k)
o Pq(xk),
PD{Cx|k+Du|kh}≥PD{CKe|kq}.
Deno ing Cj,Dj, and CK,j he j- h ows o ma ices C,
Dand CK espec i ely, and hj,q,j he j- h componen s
o hand qwe also ob ain
PD{Cjx|k+Dju|k≤hj}≥PD{CK,j e|k≤q,j}.
Gi en δq∈(0,1) and q∈(0,1), Theo em 1 p o ides a way
o ob ain {q}N−1
=0 such ha , wi h p obabili y no smalle
han 1 −δq,
PD{CK,j e|k(ζ)>q
,j}≤q,∀∈IN N−1
0,∀j∈IN nh
1.
Thus, we conclude ha , gi en δq∈(0,1) and q∈(0,1), i
{q}N−1
=0 a e ob ained acco ding o Theo em 1 hen, wi h
p obabili y no smalle han 1 −δq,
PD{Cjx|k+Dju|k≤hj}≥1−q,∀∈IN N−1
0,∀j∈IN nh
1.
We no ice ha he p e ious p obabilis ic bound does no
e e o he closed-loop beha iou , bu o he p edic ion
scheme o he s ochas ic MPC o mula ion. In he ollow-
ing sec ions, we p esen how o design a so cons ained
e sion o he con olle p oposed in (16) in o de o ob ain
closed-loop p obabilis ic gua an ees.
5. SOFT-CONSTRAINED CONTROLLER
The easibili y egion XNo p oblem Pq(·) is o en a
bounded egion a ound he o igin. Mo eo e , i he p ob-
abili y dis ibu ion o he dis u bances has no a ini e
suppo , hen he on-line ecu si e easibili y o Pq(·) can
only be gua an eed in a p obabilis ic way (see, o exam-
ple, Fleming and Cannon (2019)). In o de o ci cum en
his p oblem, we p opose a so cons ained o mula ion
o he op imiza ion p oblem ha de ines he s ochas ic
MPC con olle . The p oposed scheme elies on he no ion
o penal y unc ion (Ke igan and Maciejowski (2000)).
Gi en a penal y ac o ρ>0, and an ini ial condi ion xk,
we de ine he new op imiza ion p oblem Pρ(xk) as
min
zk, k
J(zk, k)+ρ
N−1

=0
CKz|k+D |k−h+q+
(18a)
s. . (16b),(16c),(16d),(18b)
u0|k= 0|k+Kx0|k∈U.(18c)
To gua an ee ha he con olle p o ides admissible con-
ol ac ion, we ha e explici ly added a i s s ep cons ain
on he inpu u0|kin (18c), de ining he easible i s in-
pu s o he ini e ho izon p og am. Unde e y gene al
assump ions (con ollabili y and N≥nx), p oblem Pρ(xk)
is always easible.
Mo eo e , Pρ(xk) can be cas in o he ollowing equi alen
op imiza ion p oblem using a slack a iable η
min
zk, k,η J(zk, k)+ρη1(19a)
s. . z0|k=xk,(19b)
z+1|k=AKz|k+ |k,∀∈NN−2
0,(19c)
zN−1|k=AzN−1|k+B N−1|k,(19d)
u0|k= 0|k+Kx0|k∈U,(19e)
CKz|k+D |k−h+qη, ∀∈NN−1
0(19 )
η0.(19g)
Since he componen s o ηa e es ic ed o be non-
nega i e, η1is equal o he sum o he componen s
o η. This implies ha op imiza ion p oblem (19) amoun s
o he minimiza ion o a semi-de ini e quad a ic unc ion
subjec o a se o linea equali ies and inequali ies. Thus,
(19) is a semi-de ini e quad a ic op imiza ion p oblem, o
which he e exi s many eliable sol e s sui ed o as -
embedded implemen a ions, e.g. OSQP and CVXGEN.
See S ella o e al. (2018), Ma ingley and Boyd (2012) and
e e ences he ein.
6. SAMPLED-BASED DESIGN OF THE PENALTY
FACTOR ρ
The objec i e o his sec ion is o de e mine, by means o a
sampled-based scheme, a alue o he penal y ac o ρable
o p o ide some p obabilis ic gua an ees wi h espec o
he closed-loop sa is ac ion o (2) along a gi en simula ion
ho izon M
Cxk+Dukh, k =0,...,M.
Gi ens ρ∈(0,∞) and he s a e ec o xk, he con ol
ac ion ukco esponding o he con ol scheme p esen ed
in Sec ion 5 is uk= ∗
0|k+Kxk, whe e ∗
0|kis he i s
elemen o he op imal con ol sequence ∗
k, solu ion o
(19). The esul ing con olle is de ined as uk=κ(xk,ρ),
whe e we make explici he penal y ac o ρ. Hence, he
closed-loop sys em (1) becomes
xk+1 =Axk+Bκ(xk,ρ)+ζk,∀k∈IN ≥0.
In o de o cha ac e ize he closed-loop beha iou o he
sys em, we conside a simula ion ho izon Mconside ably
la ge han he p edic ion ho izon Nused in he de ini ion
o he con olle κ(·,ρ). In pa icula , we assume ha he
ho izon Mis la ge enough o gua an ee ha he s a e xM
eaches a sa e egion a ound he e e ence s eady s a e.
Thus, he closed-loop ajec o y {xk,u
k}M
k=0 is de e mined
by
(i) he penal y ac o ρ;
(ii) he ini ial condi ion a k= 0, i.e. x0;
(iii) he unce ain ealiza ion o he dis u bances
dM={ζ0,...,ζ
M−1}.
Now, le us conside a p obabili y dis ibu ion in XN,
i.e. he easibili y egion o op imiza ion p oblem (16).
Mo eo e , gi en he simula ion ho izon M, we de ine DM
as he se o he possible alues o dMand a p obabili y
dis ibu ion in i oo. In o de o simpli y he no a ion, we
de ine Was he se o possible alues o
w={x0,dM}={x0,ζ
0,...,ζ
M−1}∈X
N×D
M=W.
We assume ha we a e able o d aw i.i.d. samples om W.
Then, he closed-loop ajec o y co esponding o con-
olle κ(·,ρ) and unce ain ealiza ion wis deno ed as
{xk(w, ρ),u
k(w, ρ)}M
k=0.
Gi en he unce ain ealiza ion wand ρ, we can de e mine
i he co esponding closed-loop ajec o y sa is ies he
con ol cons ain s by means o he compu a ion o he
ollowing pe o mance index
g(w, ρ).
=
M

k=0
Cxk(w, ρ)+Duk(w, ρ)−h+.
Hence, we ha e
g(w, ρ)=0⇔Cxk(w, ρ)+Duk(w, ρ)h, ∀k∈IN M
0,
and we can conclude ha g(w, ρ) se es as an index o
e alua e o which ex en he con ol cons ain s ha e been
sa is ied along he closed-loop ajec o y.
Now, we conside ha ρis allowed o ake alues om
a se o ini e ca dinali y Θρ={ρ1,ρ
2,...,ρ
nC}, whe e
nCis he ca dinali y o Θρ. The nex heo em p esen s
a sample-based scheme ha , o any ρ∈Θρ, p o ides a
p obabilis ic uppe bound on g(w, ρ).
Theo em 2. Le us suppose o d aw Sρi.i.d. samples o
w(i) om W=XN×D
M, i.e.
{w(1),...,w
(Sρ)},
and ha ρis a gi en disca ding pa ame e . Mo eo e ,
gi en any ρ∈Θρ, we in oduce he ollowing no a ion
γ(ρ)={g(w(i),ρ)}Sρ
i=1 ρ
.(20)
Then, wi h p obabili y no smalle han 1 −δρ, we ha e
PW{g(w, ρ)>γ(ρ)}≤ρ,∀ρ∈Θρ,(21)
p o ided ha ρ∈IN Sρ
1and
ρ−1

m=0 Sρ
mm
ρ(1 −ρ)Sρ−m≤δρ
nC
.(22)
In addi ion, (22) is sa is ied i
Sρ≥1
ρ ρ−1 + ln nC
δρ
+2( ρ−1) ln nC
δρ.(23)
P oo o Theo em 2 ollows simila de elopmen s o ha
o Theo em 1 and is omi ed o b e i y.
We no ice ha he p obabilis ic gua an ees gi en in (21)
a e alid o e e y alue o ρin Θρ. The pa icula choice
o he con olle implemen a ion depends on he speci ic
con ol applica ion. Fo example, one could choose he
alue o ρ ha minimizes γ(ρ) in Θρ. Ano he possibil-
i y is o choose he smalles ρsa is ying a p e-speci ied
cons ain on γ(ρ). The nume ical example in Mamma ella
e al. (2020) illus a es how o choose ρ. Indeed, in ha pa-
pe we demons a e he e icacy o he p oposed app oach
by means o a simple nume ical example whe e he e ec s
o he penal y ac o on he con olle a e shown, p o iding
a me hod o he use s o selec he bes alue o he penal y
ac o acco ding o he applica ion needs.

11276 Ma ina Mamma ella e al. / IFAC Pape sOnLine 53-2 (2020) 11271–11276
7. CONCLUSIONS
A s ochas ic model p edic i e con olle able o accoun o
he e ec s o addi i e s ochas ic dis u bances is p esen ed
in his pape . No es ic i e assump ions on he andom
na u e o dis u bances a e equi ed. We use a sampling
me hod o bound o line he e ec o dis u bances in a
p obabilis ic manne . A penal y based o mula ion, which
a oids in easibili y o he op imiza ion p oblem de ining
he model p edic i e con olle , is p oposed. The no el
con ol scheme mee s some gi en p obabilis ic closed-
loop speci ica ions. The equi ed sample complexi y has
a loga i hmic dependence wi h espec o he p edic ion
ho izon.
REFERENCES
Ahsanullah, M., Ne zo o , V., and Shakil, M. (2013). An
in oduc ion o O de S a is ics. A lan is P ess, Pa is.
Alamo, T., Manzano, J., and Camacho, E. (2018). Robus
design h ough p obabilis ic maximiza ion. In T. Basa
(ed.), Unce ain y in Complex Ne wo ked Sys ems. In
Hono o Robe o Tempo, 247–274. Bi kh¨ause .
Alamo, T., Tempo, R., Luque, A., and Rami ez, D. (2015).
Randomized me hods o design o unce ain sys ems:
sample complexi y and sequen ial algo i hms. Au oma -
ica, 52, 160–172.
Alamo, T., Mi asie a, V., Dabbene, F., and Lo enzen, M.
(2019). Sa e app oxima ions o chance cons ained se s
by p obabilis ic scaling. In 2019 18 h Eu opean Con ol
Con e ence (ECC), 1380–1385. IEEE.
A nold, B., Balak ishnan, N., and Naga aja, H. (1992). A
Fi s Cou se in O de S a is ics. John Wiley and Sons,
New Yo k.
Cala io e, G.C. and Fagiano, L. (2013). S ochas ic model
p edic i e con ol o LPV sys ems ia scena io op imiza-
ion. Au oma ica, 49(6), 1861–1866.
Campo, P.J. and Mo a i, M. (1987). Robus model
p edic i e con ol. In P oc. o he Ame ican Con ol
Con e ence, 1021–1026.
Fa ina, M. and Sca olini, R. (2016). Model p edic i e
con ol o linea sys ems wi h mul iplica i e unbounded
unce ain y and chance cons ain s. Au oma ica, 70, 258
– 265.
Fleming, J. and Cannon, M. (2019). S ochas ic MPC
o addi i e and mul iplica i e unce ain y using sample
app oxima ions. IEEE T ansac ions on Au oma ic Con-
ol, 64(9), 3883–3888. doi:10.1109/TAC.2018.2887054.
Hewing, L. and Zeilinge , M.N. (2018). S ochas ic model
p edic i e con ol o linea sys ems using p obabilis ic
eachable se s. In 2018 IEEE Con e ence on Decision
and Con ol (CDC), 5182–5188.
Ka g, B., Alamo, T., and Lucia, S. (2019). P obabilis ic
pe o mance alida ion o deep lea ning-based obus
NMPC con olle s. a Xi p ep in a Xi :1910.13906.
Ke igan, E.C. and Maciejowski, J.M. (2000). So con-
s ain s and exac penal y unc ions in model p edic i e
con ol. In P oceedings o UKACC In e na ional Con-
e ence on Con ol.
Limon, D., Alamo, T., Salas, F., and Camacho, E.F.
(2006). On he s abili y o cons ained MPC wi hou
e minal cons ain . IEEE ansac ions on au oma ic
con ol, 51(5), 832–836.
Limon, D., Al a ado, I., Alamo, T., and Camacho, E.F.
(2008). Mpc o acking piecewise cons an e e ences
o cons ained linea sys ems. Au oma ica, 44(9), 2382–
2387.
Lo enzen, M., Dabbene, F., Tempo, R., and Allg¨owe ,
F. (2017). S ochas ic MPC wi h o line unce ain y
sampling. Au oma ica, 81, 176–183.
Lo enzen, M., Dabbene, F., Tempo, R., and Allg¨owe , F.
(2016). Cons ain - igh ening and s abili y in s ochas ic
model p edic i e con ol. IEEE T ansac ions on Au o-
ma ic Con ol, 62(7), 3165–3177.
Mamma ella, M., Lo enzen, M., Capello, E., Pa k, H.,
Dabbene, F., Guglie i, G., Romano, M., and Allg¨owe ,
F. (2018). An o line-sampling SMPC amewo k wi h
applica ion o au onomous space maneu e s. IEEE
T ansac ions on Con ol Sys ems Technology, 1–15.
Mamma ella, M., Alamo, T., Lucia, S., and Dabbene, F.
(2020). A p obabilis ic alida ion app oach o penal y
unc ion design in s ochas ic model p edic i e con ol.
a Xi p ep in a Xi :2003.07241 1.
Ma gellos, K., Goula , P., and Lyge os, J. (2014). On
he oad be ween obus op imiza ion and he scena io
app oach o chance cons ained op imiza ion p oblems.
IEEE T ansac ions on Au oma ic Con ol, 59(8), 2258–
2263.
Ma ingley, J. and Boyd, S. (2012). CVXGEN: A code gen-
e a o o embedded con ex op imiza ion. Op imiza ion
and Enginee ing, 13(1), 1–27.
Mayne, D.Q., Rawlings, J.B., Rao, C.V., and Scokae ,
P.O. (2000). Cons ained model p edic i e con ol:
S abili y and op imali y. Au oma ica, 36(6), 789–814.
Mayne, D. and Rawlings, J. (2009). Model P edic i e
Con ol: Theo y and Design. Nob Hill Publishing.
Mesbah, A. (2016). S ochas ic model p edic i e con ol:
An o e iew and pe spec i es o u u e esea ch. IEEE
Con ol Sys ems Magazine, 36(6), 30–44.
P andini, M., Ga a i, S., and Lyge os, J. (2012). A an-
domized app oach o s ochas ic model p edic i e con-
ol. In 2012 IEEE 51s IEEE Con e ence on Decision
and Con ol (CDC), 7315–7320.
Schildbach, G., Fagiano, L., F ei, C., and Mo a i, M.
(2014). The scena io app oach o s ochas ic model
p edic i e con ol wi h bounds on closed-loop cons ain
iola ions. Au oma ica, 50(12), 3009–3018.
S ella o, B., Banjac, G., Goula , P., Bempo ad, A., and
Boyd, S. (2018). OSQP: An ope a o spli ing sol e o
quad a ic p og ams. In 2018 UKACC 12 h In e na ional
Con e ence on Con ol (CONTROL), 339–339. IEEE.
Tempo, R., Bai, E., and Dabbene, F. (1997). P obabilis ic
obus ness analysis: explici bounds o he minimum
numbe o samples. Sys ems & Con ol Le e s, 30, 237–
242.