Au oma ica 179 (2025) 112436
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Model p edic i e con ol o acking wi h implici in a ian se sI
I ene Luque a,∗, Paula Chan eu b, Daniel Limón a, José M. Maes e a
aDepa men o Sys ems and Au oma ion Enginee ing, Uni e si y o Se ille, Se ille, Spain
bDepa men o Mechanical Enginee ing, Eindho en Uni e si y o Technology, Eindho en, The Ne he lands
a i c l e i n o
A icle his o y:
Recei ed 19 June 2024
Recei ed in e ised o m 13 Feb ua y 2025
Accep ed 15 May 2025
A ailable online 19 June 2025
Keywo ds:
Model p edic i e con ol
Implici in a ian se s
T acking sys ems
a b s a c
This pape p esen s a model p edic i e con ol (MPC) echnique o acking wi h implici e minal
componen s. The con olle o mula ion includes an a i icial se poin as decision a iable, and he
e minal cons ain is de ined implici ly o an augmen ed sys em ha depends on he la e . In
his espec , ins ead o compu ing an in a ian e minal se , we conside an ex ended p edic ion
ho izon whose leng h can be bounded simply by sol ing LPs. This app oach o e comes size- ela ed
limi a ions associa ed wi h he ope a ions needed o compu ing in a ian se s, also simpli ying he
o line MPC design. The p oposed con olle is able o d i e la ge sys ems o admissible se poin s while
gua an eeing ecu si e easibili y and con e gence. Finally, he me hod is illus a ed by an academic
example, a mass–sp ing–dampe sys em o a iable-size and a mo e ealis ic case s udy o a d one.
© 2025 The Au ho (s). Published by Else ie L d. This is an open access a icle unde he CC BY-NC license
(h p://c ea i ecommons.o g/licenses/by-nc/4.0/).
1. In oduc ion
The heo e ical p ope ies o MPC, such as ecu si e easibili y
and s abili y, a e ypically gua an eed by he con enien design
o a e minal cos unc ion and a e minal posi i ely in a i-
an se , which becomes inc easingly di icul o compu e as he
size o he sys em g ows (Gilbe & Tan, 1991; Mayne, 2013,
2014). In pa icula , he explici de e mina ion o in a ian se s
is challenging e en o linea ime-in a ian sys ems due o he
equi ed compu a ions o se in e sec ions and p e-image se s.
In ac , he e exis scalable me hods ha ind app oxima ions
o hese in a ian se s, e.g., by conside ing p e-de ined polyhe-
d on shapes (T odden, 2016), ellipsoids based on linea ma ix
inequali ies (Alamo, Cepeda, & Limon, 2005), inne -ou e app ox-
ima ions (Comelli, Ola u, & Ko man, 2024), zono opes (Mo a o,
Cunha, San os, No mey-Rico, & Sename, 2021) o da a-d i en
app oaches (Be be ich, Köhle , Mülle , & Allgöwe , 2021), o
ha gene a e implici ep esen a ions o hem (Rako ić & Zhang,
2022, 2023; Wang & Junge s, 2020). Among hem, we a e pa ic-
ula ly in e es ed in he la e (Rako ić & Zhang, 2022, 2023), o
i ex ends he p edic ion ho izon gua an eeing ha he inal s a e
belongs o an in a ian se .
IThis wo k is suppo ed by he Spanish T aining P og am o Academic S a
unde G an (FPU21/05299), and by G an s PID2022-141159OB-I00 and PID2023-
152876OB-I00, unded by MCIN/AEI/10.13039/501100011033 and ERDF/EU. The
ma e ial in his pape was pa ially p esen ed a he 63 d IEEE Con e ence on
Decision and Con ol (CDC), Decembe 16–19, 2024, Milan, I aly. This pape
was ecommended o publica ion in e ised o m by Associa e Edi o Dominic
Liao-McPhe son unde he di ec ion o Edi o Flo ian Do le .
∗Co esponding au ho .
E-mail add esses: [email p o ec ed] (I. Luque), [email p o ec ed]
(P. Chan eu ), [email p o ec ed] (D. Limón), [email p o ec ed] (J.M. Maes e).
Mo eo e , some MPC o mula ions u he complica e his is-
sue, such as ha o acking (Fe amosca, Limón, Al a ado, Alamo,
& Camacho, 2009; Limón, Al a ado, Alamo, & Camacho, 2008),
which conside s an augmen ed e minal sys em — enla ging i s
dimension — o deal wi h non- ixed se poin s. Speci ically, he
MPC o acking o mula ion p esen ed in Limón e al. (2008)
and Fe amosca e al. (2009) adds an a i icial s eady s a e and
inpu as decision a iables in he op imiza ion p oblem o elax
he e minal cons ain , and ensu es ecu si e easibili y and con-
e gence o he eal se poin by using a modi ied cos unc ion.
While he amily o MPC con olle s o ack changing se poin s
is wide , e.g., Bempo ad, Casa ola, and Mosca (1997), Ga one,
Di Cai ano, and Kolmano sky (2017), Gilbe and Kolmano sky
(2002), we conside he app oach in Fe amosca e al. (2009),
Limón e al. (2008) o apply he p oposed implici me hodology
no only due o i s heo e ical gua an ees, bu also because i
enla ges he domain o a ac ion o he con olle .
The main con ibu ion o his a icle consis s in designing an
MPC o acking ha inco po a es implici e minal ing edien s,
ex ending he p elimina y wo k in oduced in Luque, Chan eu ,
Limón, and Maes e (2024). Pa icula ly, he p oposed app oach
elies on eplacing he e minal cons ain se wi h an ex ended
p edic ion ho izon o a p ede ined ini e leng h. The p esen ed
me hodology allows ob aining his leng h o a acking se ing
by sol ing linea p og ams (LPs), hus a oiding se calcula ions
and he e o e enabling i s applica ion o sys ems o any size, a
he expense o a ma ginal inc ease in he online compu a ional
bu den caused by he use o longe ho izons. As will be seen,
he use o implici in a ian se s is no s aigh o wa d in his
con ex and equi es a ailo -made adap a ion o he esul s o
egula ion ega ding he exis ence o a i icial and eal se poin s.
h ps://doi.o g/10.1016/j.au oma ica.2025.112436
0005-1098/© 2025 The Au ho (s). Published by Else ie L d. This is an open access a icle unde he CC BY-NC license (h p://c ea i ecommons.o g/licenses/by-nc/4.0/).
I. Luque, P. Chan eu , D. Limón e al. Au oma ica 179 (2025) 112436
The p oposed me hodology p o ides gua an ees o ecu si e ea-
sibili y and con e gence o he eal se poin s, which a e key
p ope ies in he use o acking MPC. Fu he mo e, his pape
discusses an al e na i e app oach o cases whe e he maximum
admissible leng h o he ex ended p edic ion ho izon is limi ed,
equi ing al e na i e s a egies o a oid he calcula ion o he
e minal egion while enabling o selec he desi ed ex ension o
he ex ended p edic ion ho izon.
The ou line o he es o he a icle is as ollows. Sec ion 2
in oduces he p elimina ies o he p oblem p esen ed. Sec ion 3
p esen s he con olle design wi h implici e minal componen s,
as well as he al e na i e app oach and heo e ical p oo s, whose
pe o mance is illus a ed in Sec ion 4 wi h h ee case s udies.
Las ly, Sec ion 5 p o ides he discussion.
No a ion. Vec o [x⊤,u⊤]⊤ is deno ed as (x,u); In and 0m×n
ep esen he iden i y and ze o ma ices o dimension n×n and
m×n, espec i ely, whe eas 0n and 1n a e column ec o s o ze os
and ones o size n×1. R, N deno e he se s o eal and na u al
numbe s, espec i ely. Likewise, gi en a,b∈N, wi h a<b,
we de ine N[a,b]:= {a,a+1, . . . , b−1,b} and Nb is gi en
o N[0,b]. Finally, [u ]T
=0 deno es ec o [u⊤
0,u⊤
1, . . . , u⊤
T]⊤ o
any gi en T∈N. The suppo unc ion h(X,·) o a closed, non-
emp y subse X∈Rn is gi en o all y∈Rn by h(X,y):=
supx{y⊤x:x∈X}.
2. P oblem se ing
In his sec ion, he sys em dynamics and he MPC o he
conside ed acking o mula ion a e in oduced.
2.1. Sys em dynamics
Conside a disc e e- ime linea sys em gi en by
xk+1=Axk+Buk,(1)
whe e xk∈Rn and uk∈Rm a e espec i ely he s a e o he
sys em and he inpu a ins an k, and ma ices A and B a e o
compa ible dimensions, i.e., A∈Rn×n and B∈Rn×m, wi h m and
n being posi i e and possibly di e en in ege s. Also, conside he
ollowing cons ain s
xk∈X,uk∈U,∀k∈N,(2)
being X⊆Rn and U⊆Rm he s a e and inpu cons ain se s,
espec i ely. Le us in oduce he ollowing assump ion:
Assump ion 1. Fo sys em (1) subjec o cons ain s (2), he
ollowing holds:
•The ma ix pai (A, B) is known and i is s ic ly s abilizable.
•Cons ain se s X and U a e con ex poly opic se s con aining
he o igin in hei in e io .
•The e exis s a eedback gain K∈Rm×n such ha ma ix
A+BK is Schu , and a posi i e de ini e ma ix P∈Rn×n
such ha
(A+BK)⊤P(A+BK)−P= −(Q+K⊤RK).(3)
The sys em pe o mance will be e alua ed h ough s age cos
unc ion
ℓ(xk,uk,xs,us)= ∥xk−xs∥2
Q+ ∥uk−us∥2
R,(4)
whe e Q∈Rn×n and R∈Rm×m a e symme ic posi i e de ini e
ma ices, and (xs,us) deno es he se poin o which we wan o
d i e he sys em. In his ega d, no ice ha any se poin o he
sys em mus sa is y he ollowing equa ion
[A−InB][xs
us]=0n.(5)
The e o e, we can pa ame e ize he pai (xs,us) h ough a i-
able θ∈Rm, i.e.,
[xs
us]=[Mθx
Mθu]
Mθ
θ, (6)
being Mθ a sui able basis o he null space o [A−InB] ha
agg ega es ma ices Mθx∈Rn×m and Mθu∈Rm×m (Fe amosca
e al., 2009).
2.2. MPC o acking wi h explici e minal componen s
The MPC o acking o mula ion conside ed in his a icle is
cha ac e ized by he ollowing (Limón e al., 2008):
(i) An a i icial se poin , say (xa
s, ua
s), is in oduced as op imiza-
ion a iables in he MPC p oblem. This a i icial se poin
will be pa ame ized by a iable θa and in oduces m new
op imiza ion a iables.
(ii) An o se cos unc ion is also added o penalize he de ia ion
o he a i icial se poin om he eal se poin s, (x
s, u
s).
(iii) An augmen ed e minal in a ian se Ψ
is used, which is
de ined o augmen ed sys em
[xk+1
θa]=[A+BK BL
0m×nIm]
Aaug
[xk
θa],(7)
whe e L= [−K Im]Mθ. In wha ollows, we will di e en i-
a e be ween he eal se poin , deno ed as (x
s, u
s), and he
a i icial se poin , i.e., (xa
s, ua
s). No e ha (7) conside s ha
sys em (1) employs con ol law
uk=ua
s+K(xk−xa
s)=Kxk+Lθa.(8)
Conside ing he abo e, he MPC o acking p oblem o be
sol ed a e e y ime ins an k adop s he ollowing o m:
V∗
N(xk,x
s)=min
u,θaVN(xk,x
s,u, θa)
s. . x0|k=xk,(9a)
xj+1|k=Axj|k+Buj|k,j∈N[0,N−1],(9b)
xj+1|k∈X,j∈N[0,N−1],(9c)
uj|k∈U,j∈N[0,N−1],(9d)
[xa
s
ua
s]=Mθθa,[xN|k
θa]∈Ψ
,(9e)
whe e N is he leng h o he p edic ion ho izon and u= [uj|k]N−1
j=0.
In his espec , subsc ip j|k indica es a p edic ion on he co e-
sponding a iable o ins an k+j made a ime k. Also, he cos
unc ion is de ined as
VN(xk,x
s,u, θa)=
N−1
∑
j=0(∥xj|k−xa
s∥2
Q+ ∥uj|k−ua
s∥2
R)
+ ∥xN|k−xa
s∥2
P+ ∥xa
s−x
s∥2
O,
(10)
whe e O∈Rn×n is a posi i e de ini e ma ix and P sa is ies (3).
No e ha , unlike MPC o egula ion, he de ia ion o he sys-
em wi h espec o he a i icial se poin is weigh ed du ing
he p edic ion ho izon, and an o se cos , i.e. ∥xa
s−x
s∥2
O, is
added o penalize he di e ence be ween he a i icial and eal
s a e e e ence. Likewise, e minal cons ain (9e) is de ined by
in a ian se Ψ
, which is compu ed o augmen ed sys em (7)
conside ing cons ain s (2), as de ailed in Limón e al. (2008).
2
I. Luque, P. Chan eu , D. Limón e al. Au oma ica 179 (2025) 112436
In pa icula , Ψ
is a polyhed al app oxima ion o he maximal
in a ian se , sa is ying
Ψ
⊆ {(x, θ)∈Rn+m:(x,Kx +Lθ)∈(X,U), θ ∈Θ},
whe e
Θ:= {θ∈Rm:Mθxθ∈X,Mθuθ∈U}.(11)
Because o he uni a y eigen alues o Aaug, se Ψ
migh no
be ini ely de e mined (Gilbe & Tan, 1991). None heless, i is
possible o scale Θ by ac o λ∈(0,1) so ha he maximal
admissible in a ian se becomes a ini ely de e mined con ex
polyhed on, say Ψ
,λ (Gilbe & Tan, 1991; Limón e al., 2008).
No e ha
Ψ
,λ ⊆ {(x, θ)∈Rn+m:(x,Kx +Lθ)∈(X,U), θ ∈λΘ},(12)
and
[Axk+B(Kxk+Lθa)
θa]∈Ψ
,λ o all [xk
θa]∈Ψ
,λ.
As de ailed in Limón e al. (2008) and Fe amosca e al. (2009),
he acking o mula ion (9) allows d i ing he sys em s a e o
any admissible a ge se poin . Howe e , i equi es compu ing
in a ian se Ψ
,λ o augmen ed dynamics (7), whose dimension
is n+m. While he e minal cos is simple o calcula e, he
cons uc ion o such a e minal se becomes in ac able o la ge
sys ems. To sol e his, his a icle e o mula es MPC p oblem (9)
using implici e minal componen s, i.e., a oiding he need o
explici ly cha ac e izing Ψ
,λ.
3. MPC o acking wi h implici e minal componen s
In wha ollows, we ex end esul s on implici e minal com-
ponen s de i ed o egula ion in Rako ić and Zhang (2023) o
acking p oblems.
3.1. Implici e minal se o egula ion
Fo he egula ion p oblem, ha is (x
s,u
s)=0n+m, he e mi-
nal con ol law can be simply de ined as uk=Kxk, and, he e o e,
he e minal dynamics a e gi en by
xk+1=(A+BK)xk,(13)
which is also ob ained i we ix he a i icial and eal se poin o
he o igin in (7) and (8). Also, gi en (2), he cons ain s o he
e minal s age can be compac ly de ined as
xk∈X := {x∈Rn:x∈X,Kx ∈U}.(14)
Conside ing he abo e, le us in oduce he ollowing heo em,
which es ablishes a su icien condi ion o he exis ence o he
maximal posi i ely in a ian se . Recall ha a se Ω⊂Rn is
de ined as a posi i ely in a ian se o cons ain s x∈X i and
only i (Ke igan, 2001; Rako ić & Zhang, 2022):
xk∈Ω⇒ ∃uk∈U such ha xk+1∈Ω,xk+1∈X.
Then, a se is de ined as he maximal posi i ely in a ian se i
i is posi i ely in a ian and con ains all he posi i ely in a ian
se s in Ω (Ke igan, 2001).
Theo em 1 (Rako ić and Zhang (2023, Theo em 1 and 2)). Suppose
Assump ion 1 holds. Then, he maximal posi i ely in a ian se o
sys em (13) and cons ain s (14) is ini ely de e mined i and only i
o some M∈N some o he ollowing holds
M
⋂
j=0
(A+BK)−jX ⊆(A+BK)−(M+1)X
o
X ⊆(A+BK)−(M+1)X .
(15)
Gi en Theo em 1, he maximal posi i ely in a ian se o
sys em (13) and cons ain s (14), say Ψ , is a nonemp y closed
polyhed al se con aining he o igin in i s in e io , de ined as
Ψ =
M
⋂
j=0
(A+BK)−jX .(16)
An al e na i e app oach o check whe he a gi en s a e belongs
o Ψ can be in oduced using a ajec o y o leng h M (Rako ić &
Zhang, 2023). Tha is, xN∈Ψ i sequence [xj]N+M
j=N is such ha
∀j∈N[N,N+M],xj∈X ,
∀j∈N[N,N+M−1],xj+1=(A+BK)xj,(17)
whe e M,N∈N, wi h M≥1 sa is ying (15). No e ha i M=0,
hen Ψ =X. The la e se es as he basis o he implici
e o mula ion o he e minal componen s.
Le se X ⊆Rn ( ecall (14)) be a closed polyhed al se whose
i educible ep esen a ion is
X := {x∈Rn:(C+DK)x≤1p},(18)
whe e ma ix pai (C,D)∈Rp×n×Rp×m is de ined acco d-
ingly. No e ha p deno es he numbe o inequali ies de ining X .
Gi en (18), X can be simila ly de ined as
X = {x∈Rn: ∀i∈N[1,p],X⊤
ix≤1},(19)
whe e X⊤
i is he i h ow o he ma ix (C+DK). Following Rako ić
and Zhang (2023), we ha e ha (15) holds ue i and only i one
o he ollowing holds o all i∈N[1,p]:
h(
M
⋂
j=0
(A+BK)−jX ,((A+BK)M+1)⊤Xi)≤1
o
h(X ,((A+BK)M+1)⊤Xi)≤1.
(20)
The le -hand sides o he inequali ies in (20) can be calcula ed
o di e en i h ough he ollowing compu a ionally simple LPs,
espec i ely:
sup
(x,[xj]M
j=0){X⊤
i(A+BK)M+1x:xj=(A+BK)jx,xj∈X ,∀j∈NM}
o
sup
x
{X⊤
i(A+BK)M+1x:x∈X }.
Then, gi en any in ege M, e i ying (15) educes o sol ing p LPs.
In addi ion, he sea ch o such an in ege M can be pe o med by
sea ching h ough any sui ably gene a ed sequence o posi i e in-
c easing in ege s. See Rako ić and Zhang (2022, 2023) o u he
de ails.
3.2. Implici e minal se o acking
Assume ha sys em (1) is con olled by con ol law (8). Then,
conside ing he augmen ed dynamics, he ollowing holds o a
gi en θ:
[xk+1
θ]=Aaug [xk
θ],(21)
whe e he cons ain s o his e minal augmen ed dynamics a e
de ined as ollows
Xaug, := {(x, θ)∈Rn+m:x∈X,Kx +Lθ∈U, θ ∈λΘ}.(22)
Simila ly o (18), le se Xaug, ⊆Rn+m be a closed polyhed on
ha can be de ined i educibly as
Xaug, := {(x, θ)∈Rn+m:˜
G[x
θ]≤1˜
p},(23)
3
I. Luque, P. Chan eu , D. Limón e al. Au oma ica 179 (2025) 112436
whe e
˜
G=[˜
C+˜
DK ˜
DL
0˜
W]∈R˜
p×(n+m).(24)
Also, ˜
C=C,˜
D=D,˜
W=(CMθx+DMθu)/λ and 0 is a ma ix o
ze os o he app op ia e size. Remembe ha ˜
p is de ined as he
numbe o inequali ies in Xaug, . Gi en (23), se Xaug, can also be
de ined as
Xaug, ={(x, θ)∈Rn+m: ∀i∈N[1,˜
p],X⊤
aug,i[x
θ]≤1},(25)
whe e X⊤
aug,i is he i h ow o he ma ix ˜
G. In addi ion, le us
conside some ˜
M∈N sa is ying a condi ion simila o (15) bu
adap ed o he augmen ed sys em, i.e.,
˜
M
⋂
j=0
A−j
augXaug, ⊆A−(˜
M+1)
aug Xaug,
o
Xaug, ⊆A−(˜
M+1)
aug Xaug, ,
(26)
and he ollowing e minal se
Ψ
,λ =
˜
M
⋂
j=0
A−j
aug [X
λΘ].(27)
Gi en ha , o some λ∈(0,1), he augmen ed e minal se o
acking Ψ
,λ is ini ely de e mined (Gilbe & Tan, 1991; Limón
e al., 2008), i is possible o ensu e ha he e exis s a ini e
alue o ˜
M ha sa is ies Eq. (26) (see also Rako ić & Zhang, 2022,
Co olla y 4).
Theo em 2. Suppose Assump ion 1 holds, and conside some
˜
M,N∈N, wi h ˜
M≥1 sa is ying (26). Then, cons ain (xN, θ)∈
Ψ
,λ is sa is ied i and only i he e exis s a sequence [xj]N+˜
M
j=N such
ha
∀j∈N[N,N+˜
M],(xj, θ)∈Xaug, ,
∀j∈N[N,N+˜
M−1],xj+1=(A+BK)xj+BLθ. (28)
To ul ill (26), one o he ollowing condi ions mus hold,
espec i ely, o all i∈N[1,˜
p]:
h(
˜
M
⋂
j=0
A−j
augXaug, ,(A˜
M+1
aug )⊤·Xaug,i)≤1
o
h(Xaug, ,(A˜
M+1
aug )⊤·Xaug,i)≤1.
(29)
The ollowing compu a ionally simple LPs allow us o check he
inequali ies in (29) o any i∈ [1,˜
p]:
sup
(x,θ,[xj]˜
M
j=0,[θj]˜
M
j=0){X⊤
aug,i·A˜
M+1
aug [x
θ]:
[xj
θj]=Aj
aug [x
θ]∈Xaug, ∀j∈N[0,˜
M]}
o
sup
(x,θ)
{X⊤
aug,i·A˜
M+1
aug [x
θ]:(x, θ)∈Xaug, }.
As a esul , a new pa ame e ˜
M sa is ying (26) can be ound by
sea ching h ough posi i e inc easing in ege s and sol ing ˜
p LPs
o each o hem.
3.3. MPC o acking wi h implici e minal componen s
The p oposed MPC, including acking and implici e minal
cons ain s, is de ailed in his subsec ion. Following Theo em 2,
he p edic ion ho izon is pa i ioned in wo s ages: a i s pa o
leng h N, and a e minal pa o leng h ˜
M. Tha is, sequences o
leng h N+˜
M will be compu ed.
Based on (9), he p oposed op imal con ol p oblem o be
sol ed a e e y ime s ep is de ined as
min
u,θaVN+˜
M(xk,x
s,u, θa)
s. . x0|k=xk,(30a)
xj+1|k=Axj|k+Buj|k,j∈N[0,N+˜
M−1],(30b)
xj+1|k∈X,j∈N[0,N+˜
M−1],(30c)
uj|k∈U,j∈N[0,N+˜
M−1],(30d)
uj|k=Kxj|k+Lθa,j∈N[N,N+˜
M−1],(30e)
[xa
s
ua
s]=Mθθa(30 )
θa∈λΘ.(30g)
He e, cons ain (30e) imposes he e minal con ol law and is
only conside ed du ing he second pa o he p edic ion ho izon,
p e iously called he e minal s age. Also, he objec i e unc ion
akes in o accoun he pe o mance up o p edic ion ime in-
s an N+˜
M plus he e minal and he o se cos s, as i was
de ined in (10). Finally, no e ha he ex ended p edic ion ho izon
eplaces he explici e minal cons ain (9e).
Rema k 1. The implici app oach a oids compu ing o line he
explici ep esen a ion o he in a ian se in exchange o ex-
ending he p edic ion ho izon, which implies inc easing he on-
line compu a ional bu den. Howe e , since QPs can be sol ed in
polynomial ime, his may no be a limi ing ac o in mos eal
applica ions. Also, no ice ha once ˜
M is known, he maximal
in a ian se in closed- o m can also be ound using (27), a oiding
he need o check con e gence condi ions a e e y i e a ion.
Rema k 2. An implici e minal cos unc ion o acking can be
de ined as ollows, in e ed om Rako ić and Zhang (2023):
V
F,imp(xN|k,xa
s,ua
s)=(1 −ϵ)−1
N+˜
M−1
∑
j=N
(∥xj|k−xa
s∥2
Q+ ∥uj|k−ua
s∥2
R),
whe e ϵ∈ [0,1) is minimized so ha he implici bound in
he e minal cos becomes igh . Al hough his is a alid op ion,
compu ing P as conside ed in (10) is ypically no expensi e and
p o ides and mo e accu a e es ima ion o he cos - o-go. Fo his
eason, i is chosen in his pape .
3.4. Theo e ical p ope ies
He ea e , we p o e ha he ini ial easibili y o op imiza-
ion p oblem (30) also implies ecu si e easibili y. In addi ion,
con e gence o he eal se poin s, i admissible, is also p o en.
Theo em 3. Assume ha a ins an k he e exis s a solu ion (u∗
k, θ∗
k)
o p oblem (30). Then, we can ind a easible solu ion o (30) a all
ins an s ≥k.
P oo . Le xk be he sys em s a e a ins an k, and conside solu-
ion (u∗
k, θ∗
k). No ice ha sequence u∗
k, and i s associa ed p edic ed
4
I. Luque, P. Chan eu , D. Limón e al. Au oma ica 179 (2025) 112436
s a e sequence a e gi en by
u∗
k=(u∗
0|k,u∗
1|k, . . . , u∗
N|k, . . . , u∗
N+˜
M−1|k),
x∗
k=(x∗
0|k,x∗
1|k, . . . , x∗
N|k, . . . , x∗
N+˜
M|k),
whe e x∗
0|k=xk∈X. By cons uc ion, i ollows ha
∀j∈N[0,N+˜
M−1],x∗
j+1|k=Ax∗
j|k+Bu∗
j|k,(31a)
∀j∈N[N,N+˜
M−1],u∗
j|k=Kx∗
j|k+Lθ∗
k,(31b)
∀j∈N[0,N+˜
M−1],x∗
j+1|k∈X,u∗
j|k∈U.(31c)
Le us de ine a candida e solu ion (˜
uk+1,˜
θk+1) o p oblem (30)
a ins an k+1. In pa icula , conside ˜
θk+1=θ∗
k and
˜
uk+1= [˜
uj|k]N+˜
M−1
j=0=[[u∗
j|k]N+˜
M−1
j=1
Kx∗
N+˜
M|k+Lθ∗
k],
˜
xk+1= [˜
xj|k]N+˜
M
j=0=[[x∗
j|k]N+˜
M
j=1
Ax∗
N+˜
M|k+BKx∗
N+˜
M|k+BLθ∗
k],
(32)
whe e ˜
xk+1 is he associa ed p edic ed s a e sequence. No e ha ,
gi en x∗
0|k=xk, and conside ing (31a), we ha e ha xk+1=
Axk+Bu∗
0|k=x∗
1|k. By cons uc ion, we ha e ha ˜
xj|k+1∈X and
˜
uj|k+1∈U o all j∈N[0,N−2]. Likewise,
˜
xN−1|k+1=x∗
N|k∈X ⊆X,
˜
uN−1|k+1=Kx∗
N|k+Lθ∗
k∈U.
I is also known ha cons ain ˜
xN|k+1∈X is hen implici ly
applied gi en (16) and (32). Hence, du ing he e minal s age,
whe e j∈N[N,N+˜
M−1], i holds ha
˜
xj+1|k+1∈X ⊆X,
˜
uj|k+1=K˜
xj|k+1+L˜
θk+1∈U.(33)
The e o e, candida e solu ion (˜
uk+1,˜
θk+1) is a easible solu ion
o p oblem (30) a ins an k+1. By induc ion, ecu si e easibili y
is gua an eed o all ime ins an s. ■
Theo em 4. Le x0∈XN+˜
M, wi h XN+˜
M being he domain o
a ac ion o con olle (30). Then, s a e xk o he sys em con olled
by (30) will con e ge o x
s, i admissible, as k ends o in ini y.
P oo . No e ha XN+˜
M ep esen s he se o s a es o which
op imiza ion p oblem (30) is easible. F om he abo e, i ollows
ha i xk∈XN+˜
M, hen s a e xk+1 sa is ies xk+1∈XN+˜
M.
Consequen ly, XN+˜
M is a posi i ely in a ian se o he closed-
loop sys em. Likewise, gi en ha X is bounded, se XN+˜
M is also
bounded, he eby implying s abili y o he sys em. Below, we
demons a e con e gence by e i ying ha he op imal cos is
a Lyapuno unc ion o he closed-loop sys em, and ha he
chosen a i icial se poin con e ges o he eal one.
Le VN+˜
M(xk+1,x
s,˜
uk+1,˜
θk+1) be he cos a ins an k+1 as-
socia ed wi h candida e solu ion (˜
uk+1,˜
θk+1) (see (32)). Also,
no a ion zj|k=(xj|k,uj|k) is in oduced o cla i y. Then, we ha e
ha :
VN+˜
M(xk+1,x
s,˜
uk+1,˜
θk+1)−VN+˜
M(xk,x
s,u∗
k, θ∗
k)=
ℓ(˜
zk+N+˜
M|k+1,˜
θk+1)−ℓ(z∗
k|k, θ∗
k)+VF(˜
zk+N+˜
M+1|k+1,˜
θk+1)−
VF(z∗
k+˜
M+N|k, θ∗
k),
whe e ℓ(·) deno es he s age cos unc ion as in (4), and VF(·)
is he e minal cos in (30) ( ecall (10)). Then, gi en (33), i is
possible o s a e ha
VN+˜
M(xk+1,x
s,˜
uk+1,˜
θk+1)−VN+˜
M(xk,x
s,u∗
k, θ∗
k)≤ −ℓ(z∗
k|k, θ∗
k).
By applying he p inciple o op imali y, we ha e ha :
VN+˜
M(xk+1,x
s,u∗
k+1, θ∗
k+1)−VN+˜
M(xk,x
s,u∗
k, θ∗
k)≤ −ℓ(z∗
k|k, θ∗
k).
F om his, i ollows ha he op imal cos is s ic ly dec easing
and p o ides a Lyapuno unc ion o he sys em. Gi en his, we
in e ha limk→∞ ∥xk−xa,∗
s,k∥Q=0. No e ha xa,∗
s,k=Mθxθ∗
k
is he chosen a i icial s a e a ins an k. Finally, in Limón e al.
(2008, Lemma 3) and Fe amosca e al. (2009, Lemma 2), i is
p o ed by con adic ion ha i xk=xa,∗
s,k, hen ∥xk−x
s∥O=0.
Consequen ly, con e gence o he sys em s a e o x
s, i admissible,
is demons a ed. ■
3.5. Al e na i e implici design o acking
This sec ion in oduces an al e na i e app oach o he design
o s abilizing p edic i e con olle s wi hou e minal cons ain
ollowing Limón, Fe amosca, Al a ado, and Alamo (2018). This
al e na i e is pa icula ly ele an o scena ios whe e ex ending
he p edic ion ho izon by a leng h ˜
M is no easible o desi ed.
Addi ionally, o such sys ems, he explici compu a ion o he
in a ian se may also p o e o be compu a ionally in ac able.
Le us conside a e minal cos unc ion simila o he one
in (10), i.e. VF(xN+¯
M|k, θa)= ∥xN+¯
M|k−xa
s∥2
P, as well as he
e minal con ol law de ined in (8). I should be no ed ha he
new a iable ¯
M is di e en om M and ˜
M, and can be selec ed
o each sys em based on he desi ed pe o mance o any o he
speci ic equi emen s. Also, choose some scala α > 0 and de ine
se :
Ψα= {(x, θ)∈Rn+m:VF(x, θ)≤α},(34)
such ha Ψα is an in a ian se o acking. Finally, de ine he
ollowing MPC p oblem, which is simila o (30) bu conside s an
use -de ined ho izon ex ension and adds a new pa ame e , γ, in
he objec i e unc ion:
min
u,θaVγ
N+¯
M(xk,x
s,u, θa)
s. . x0|k=xk,(35a)
xj+1|k=Axj|k+Buj|k,j∈N[0,N+¯
M−1],(35b)
xj+1|k∈X,j∈N[0,N+¯
M−1],(35c)
uj|k∈U,j∈N[0,N+¯
M−1],(35d)
uj|k=Kxj|k+Lθa,j∈N[N,N+¯
M−1],(35e)
[xa
s
ua
s]=Mθθa(35 )
θa∈λΘ.(35g)
Pa icula ly, Vγ
N,¯
M(xk,x
s,u, θa) ep esen s he cos unc ion wi h
he e minal cos scaled by γ, ha is, γVF(xN+¯
M|k, θa).
Fo he con olle abo e, wi h γ≥1, ecu si e easibil-
i y and con e gence o admissible se poin s a e ensu ed o
all x∈Υ¯
M,γ (x
s) (Limón e al. (2018, Theo em 3)), wi h Υ¯
M,γ (x
s)
being de ined as:
Υ¯
M,γ (x
s)= {x∈Rn:Vγ∗
N+¯
M(x,x
s)−V∗
O(x,x
s)≤(N+¯
M)d+γ α},
whe e Vγ∗
N+¯
M(x,x
s) is he op imal alue o he objec i e unc ion
in (35), V∗
O(x,x
s) is he associa ed o se cos , and d ep esen s
a posi i e scala such ha ℓ(x,u,xs,us)≥d o all (x, θ)/∈Ψα.
No ice ha egion Υ¯
M,γ (x
s) is enla ged as ¯
M o γ inc ease. Finally,
i is wo h men ioning ha his con olle design only equi es
selec ing he alues o ¯
M, γ, and α, a oiding any explici o
implici es ima ion o he e minal in a ian se .
5
I. Luque, P. Chan eu , D. Limón e al. Au oma ica 179 (2025) 112436
Table 1
Cumula i e cos s o di e en alues o λ.
λAcademic example D one
˜
MCumula i e cos (×106)˜
MCumula i e cos (×106)
0.99 10 2.0067 203 3.1548
0.89 5 2.4813 88 3.1548
0.79 4 3.0960 64 3.1548
0.69 3 3.8597 50 11.5174
0.59 3 4.7774 39 44.8102
0.49 3 5.8491 31 112.0422
0.39 2 7.0748 25 271.2111
Fig. 1. S a e e olu ion on plane (x1,x2).
4. Examples
The pe o mance o he con olle will be illus a ed in h ee
examples o di e en dimensions and dynamics, simula ed using
YALMIP wi h sol e GUROBI (Lö be g, 2004).
4.1. Academic example
The i s example is a low-dimensional sys em, whose dy-
namics a e de ined by ma ices A and B in Limón e al. (2008).
The sys em is cons ained by ∥xk∥∞≤3 and ∥uk∥∞≤2, o
all k≥0. Fo he con olle design, weigh ing ma ices Q=10 I2,
R=100 I2 and O=10.000 I2 a e used; N is se o 10; and gain K
is ob ained om he disc e e LQR solu ion. Likewise, he alue
o ˜
M ha sa is y he p esen ed condi ions o λ=0.9 is ˜
M=5.
The s a e ajec o y on he plane (x1,x2) is shown in Fig. 1,
wi h x1 and x2 being he wo componen s o he sys em s a e.
We ha e conside ed di e en se poin s ( ep esen ed wi h g een
do s), wi h he las o hem being no admissible. The in a ian
se o egula ion is also shown in Fig. 1, whe e i can be seen
ha i is con ained in o he p ojec ion o ou augmen ed in a ian
se Ψ
,λ on o he plane (x1,x2).
Also, Table 1 p esen s a compa ison o he cumula i e pe o -
mance cos o di e en alues o λ. Speci ically, he cumula i e
pe o mance cos s a e compu ed using he ollowing pe o mance
indica o :
Vcc =
Tsim
∑
k=0
(∥xk−xa
s,k∥2
Q+ ∥uk−ua
s,k∥2
R)+
Tsim
∑
k=0
∥xa
s,k−x
s∥2
O,
whe e (xa
s,k,ua
s,k) deno es he a i icial se poin compu ed a in-
s an k and Tsim is used o deno e he numbe o simula ed ime
ins an s (210 in his example).
I is clea ha he cumula i e cos inc eases as λ dec eases.
This is expec ed since λ scales he se Θ, hus es ic ing he
admissible alues o θa. Likewise, an in e es ing p ope y is
in e ed om Table 1: ˜
M dec eases wi h he educ ion o λ,
o his causes he eachable se poin s o mo e away om he
cons ain s. This howe e may esul in ce ain equilib ium poin s
o he sys em no being eached. Al hough, in e ms o pe o -
mance, he mos con enien op ion is o choose λ close o 1,
his obse a ion p o ides a new deg ee o eedom. Speci ically, i
we de e mine he minimum λ equi ed o pa ame e ize he eal
se poin s o in e es o he sys em, we can po en ially educe he
necessa y ˜
M.
The ime equi ed o compu e o line he e minal ho izon
leng h ˜
M was 0.5874s, whe eas he ime o explici ly compu e he
maximal posi i ely in a ian se was 0.6315s. As o he online
con ol, he a e age ime o sol e p oblem (30) was 0.0047s,
whe eas o sol e (9) we needed 0.0044s on a e age. Recall ha (9)
e e s o he con ol p oblem wi hou he ex ended p edic ion
ho izon and an explici e minal se . While he simplici y o his
academic example does no allow o show signi ican compu a-
ional bene i s, as i migh easonably be expec ed, i demon-
s a es he p oposed design’s sui abili y o acking pu poses
while opening up he possibili y o scaling o mo e complex
sys ems, whe e compu ing in a ian se s is mo e challenging.
Finally, he al e na i e design p oposed in Sec ion 3.5 has been
implemen ed, and some key esul s a e p esen ed in Fig. 2. To
illus a e he bene i s o his me hod, we selec ed a alue o ¯
M
smalle han ˜
M=5. Speci ically, we conside ¯
M=2, γ=20,
and α=25. Wi h hese pa ame e s, he esul ing alue o d
was 5.6322. Fig. 2 shows he s abili y egion Υ¯
M,γ (x
s) o he
admissible a ge s a e x
s= [−0.5688,−0.0523]⊤. This indica es
ha i he ini ial s a e o he sys em lies wi hin his egion, he
MPC o acking wi hou e minal cons ain 3.5 will asymp o -
ically s abilize he sys em. No ably, se Υ¯
M,γ (x
s) is p ac ically as
la ge as he p ojec ion o in a ian se Ψ
,λ.
4.2. Mass–sp ing–dampe sys em o inc easing size
This subsec ion applies he p oposed MPC o a modi ied e -
sion o he sys em in Ri e so and Fe a i-T eca e (2012), T odden
and Maes e (2017). I consis s on se e al ca s connec ed by a
sp ing–dampe s uc u e, as shown in Chan eu , Maes e, Fe -
amosca, Mu os, and Camacho (2021). The dynamics o each ca i
a e modeled by:
[˙
i
˙ i]=[0 1
−1
mikij −1
mihij]
Aii
[ i
i]+[0
50]ui+wi,(36)
wi=∑
j∈Ni[0 0
1
mikij 1
mihij]
Aij
[ j
j],(37)
6
I. Luque, P. Chan eu , D. Limón e al. Au oma ica 179 (2025) 112436
Table 2
Compu a ion imes o di e en numbe s o ca s. O line imes e e o he compu a ion o he maximal RPI o he calcula ion o ˜
M o he explici and implici
cases, espec i ely. Online imes e e o he a e age ime o each MPC op imiza ion.
Numbe o ca s
3 5 10 15 25 30 50 70 100 200
O line explici ime (s) 7.6293 15.5986 54.4716 153.2055 541.9198 799.4687 3.0095×1038.7399×1036.9640×104–
O line implici ime (s) 2.7063 3.7123 6.8664 21.5659 52.3099 73.6518 201.9882 390.5777 881.0654 4.6101×103
Online explici ime (s) 0.0085 0.0107 0.0761 0.1170 0.2924 0.3138 0.2052 0.2010 0.3200 –
Online implici ime (s) 0.0095 0.0189 0.1122 0.1256 0.1582 0.1904 0.3101 0.4831 0.7917 2.0693
Fig. 2. S abili y egion eached by applying he al e na i e design me hod.
whe e he s a e is de ined by he displacemen o ca i om
an equilib ium posi ion, i, and i s eloci y, i; and inpu ui
ep esen s a o ce ha can be applied on ca i. The e exis
coupling e ms be ween adjacen ca s due o he sp ings and
dampe s ha binds hem oge he . In his espec , kij =kji
and hij =hji ep esen he sp ing s i nesses ([N/m]) and damping
ac o s ([N/(m×s)]), and mi is he mass ([kg]) o ca i. The
con inuous- ime dynamics a e disc e ized using ze o-o de hold
and a sampling ime o 0.02s.
We simula ed di e en sys em sizes by p og essi ely inc eas-
ing he numbe o ca s, say Nca s, om 3 o 200. In all cases,
he objec i e was o con ol all ca s owa ds a a ge se poin
while sa is ying he ollowing cons ain s in s a e and con ol
inpu : | i| ≤ 4, | i| ≤ 1 and |ui|<1, o all i∈ {1,2, . . . , Nca s}.
The weigh ing ma ices we e Qi= [1 0;0 1.5] and Ri=20, o
all i∈ {1,2, . . . , Nca s}, and λ=0.9.
As can be seen in Table 2, he ime o explici ly compu e o line
he in a ian se o he sys em inc eases signi ican ly, e en ually
becoming non- iable. In con as , he ime o compu e ˜
M wi h
he p oposed implici me hod emains ac able in all cases. I is
impo an o no e ha he o line compu a ion ime was limi ed
o a maximum o 48 h (2 days). Also, he online compu a ion
imes a e calcula ed as he a e age ime o sol e (30) o e he
en i e simula ion o each numbe o ca s, cap u ing he o e all
inc easing end. The di e ence in he online imes be ween he
wo me hods inc eases p og essi ely as he size o he sys em
does, al hough i ne e becomes a limi ing ac o o he p oposed
app oach. Fo ins ance, o a sys em comp ising 100 ca s, he
o line compu a ion ime o he explici me hod is 79 imes
highe han ha o he analogous measu e o he p oposed
me hod, whe eas he online compu a ion ime o ou me hod
is only 2.5 imes highe han o he explici app oach, as shown
in Table 2.
Fig. 3. Quad o o posi ion ajec o y along he simula ion.
4.3. D one
The model o a d one wi h 12 s a e a iables desc ibed in
Bea d (2008), Romagnoli, K ogh, de Niz, H is ozo , and Sinop-
oli (2023) is employed now o illus a e he applicabili y o
he p oposed MPC me hod o eal-wo ld sys ems. Le us b ie ly
indica e ha he s a e o he sys em agg ega es he posi ion
in [m] and linea eloci ies in [m/s] in a h ee dimensional space,
say (px,py,pz) and ( x, y, z), espec i ely; and also he angles
oll, pi ch and yaw, in [ ad], and hei co esponding angula
eloci ies, in [ ad/s]. Likewise, u con ains he h us , exp essed
in [N], and he o ques τφ,µ,ψ in [N· m], which a e associa ed
wi h he oll, pi ch, and yaw. The s a e cons ain s a e he ones
in Romagnoli e al. (2023), while he ollowing inpu cons ain s
a e conside ed: |F| ≤ 1.5,|τφ,µ,ψ | ≤ 0.043.
The cos unc ion is de ined by Q=10 I12,R=100 I4,
O=106I12, and N=5. The e minal eedback gain, K, is
ob ained as he solu ion o he disc e e LQR and λ=0.9. The
simula ion leng h is o 900 ime s eps wi h Ts=0.04s being he
sampling pe iod, esul ing in a simula ion o 36s. The e minal
ho izon leng h ˜
M ha sa is ies condi ion (26) is ˜
M=92.
Tes s we e ca ied ou conside ing a linea model o he
quad o o . Fig. 3 shows he esul ing posi ion ajec o y, oge he
wi h he co esponding a i icial and eal se poin s o he po-
si ion. I can be seen ha he quad o o is able o each all o
hem. The eal a ge se poin s ha e been selec ed by inding
admissible equilib ium poin s o he sys em h ough ma ix Mθ.
These se poin s only in ol e non-ze o s a e alues in posi ion and
in yaw angle due o he dynamics o he sys em. Again, in Table
1 i can be clea ly seen how dec easing λ makes i much mo e
di icul o he sys em o each he eal se poin s and, he e o e,
he cumula i e cos inc eases.
Rega ding compu a ion ime, inding ˜
M ook 16.9558s,
whe eas explici ly compu ing he maximum in a ian se e-
qui ed 137.1241s. Tha is, i was possible o achie e a educ ion
o 87.63% in he o line compu a ion ime, as inding ˜
M educes
o sol ing simple LPs, while ob aining he explici in a ian se
7
I. Luque, P. Chan eu , D. Limón e al. Au oma ica 179 (2025) 112436
wi h adi ional me hods equi es i e a i e se ope a ions. The
p oposed d one sys em has been chosen as a limi -case example,
whe e he explici compu a ion is s ill easible hough compu-
a ionally expensi e. As expec ed, he online compu a ion ime
inc eases sligh ly wi h he p oposed me hod due o he ex ended
p edic ion ho izon and he esul ing accumula ion o cons ain s.
Howe e , in ou simula ions, his inc ease was ma ginal: sol ing
he p oposed MPC p oblem (30) equi ed 0.0164s on a e age,
whe eas sol ing (9) ook 0.0114s on a e age. Finally, no ice ha ,
i implemen ed in a eal-wo ld se ing, he compu a ion imes
could be u he dec eased by employing a as e p og amming
language.
5. Conclusions
An MPC o mula ion o acking based on Fe amosca e al.
(2009), Limón e al. (2008) wi h implici e minal componen s
has been p esen ed. I a oids he need o he explici cha ac-
e iza ion o he maximal posi i ely in a ian se o he sys em
o de ine he e minal cons ain and uses ins ead an ex ended
p edic ion ho izon. The p oposed con olle can be e icien ly
designed and s ill bene i s om he addi ion o a i icial a iables
ha cha ac e ize he acking o mula ion. In his way, i o e s an
al e na i e ha can handle la ge sys ems in a ac able manne .
I has been shown ha he o line cos o he con olle design
is signi ican ly educed o la ge sys ems, while he online com-
pu a ion imes inc ease sligh ly. Finally, ecu si e easibili y and
con e gence p ope ies ha e been p o ed. As a line o u u e
esea ch, he applica ion o he p oposed app oach o nonlinea
sys ems and mo e gene al cons ain s will be conside ed, e.g., a
na u al ex ension applies o cons ain s se s ha a e no only
polyhed al bu also ellipsoidal, o in e sec ions o bo h.
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I ene Luque ecei ed he B.S. deg ee in Indus ial
Enginee ing in 2020 and he M.S. deg ee in Robo ics
and Au oma ion Enginee ing in 2022, bo h om he
Uni e si y o Se ille, Spain. She is cu en ly pu suing
he Ph.D. deg ee in Au oma ion Enginee ing a he
same ins i u ion unde he Spanish Uni e si y P o-
esso T aining P og am (FPU). She wo ked o ERC
Ad anced G an OCONTSOLAR be ween 2021 and 2022.
He esea ch ocuses on model p edic i e con ol and
cybe secu i y app oaches o cybe –physical sys ems.
Paula Chan eu is an Assis an P o esso a he
Depa men o Mechanical Enginee ing o Eindho en
Uni e si y o Technology (The Ne he lands). She e-
cei ed he Ph.D. deg ee in Au oma ion Enginee ing
om he Uni e si y o Se ille (Spain) in 2022, whe e
she was a p edoc o al ellow unde he Spanish Uni e -
si y P o esso T aining P og am (FPU). Be ween 2022
and 2023, she wo ked o ERC Ad anced G an OCON-
TSOLAR. He esea ch is amed wi hin he ield o MPC,
wi h emphasis on i s noncen alized implemen a ions.
Daniel Limón ecei ed he M.Eng. and Ph.D. deg ees
in elec ical enginee ing om he Uni e si y o Se ille,
Se ille, Spain, in 1996 and 2002, espec i ely. F om
1999 o 2007, he was an Assis an P o esso wi h he
Depa amen o de Ingenie ía de Sis emas y Au omá ica,
Uni e si y o Se ille, om 2007 o 2017 Associa e
P o esso and since 2017, Full P o esso in he same
Depa men . He has been isi ing esea che a he
Uni e si y o Camb idge and he Mi subishi Elec ic
Resea ch Labs in 2016 and 2018 espec i ely. D . Limon
has been a Keyno e Speake a he In e na ional Wo k-
shop on Assessmen and Fu u e Di ec ions o Nonlinea Model P edic i e Con ol
in 2008 and Semiplena y Lec u e a he IFAC Con e ence on Nonlinea Model
P edic i e Con ol in 2012. He has been he Chai o he i h IFAC Con e ence on
Nonlinea Model P edic i e Con ol (2015). His cu en esea ch in e es s include
model p edic i e con ol, s abili y and obus ness analysis, acking con ol and
da a-based con ol wi h applica ion o e icien ope a ion o buildings and wa e
dis ibu ion ne wo ks and spacec a ende ouz s a egies.
8
I. Luque, P. Chan eu , D. Limón e al. Au oma ica 179 (2025) 112436
José M. Maes e holds a PhD om he Uni e si y o
Se ille, whe e he cu en ly se es as a ull p o esso . He
has held posi ions a TU Del , he Uni e si y o Pa ia,
Kyo o Uni e si y, and he Tokyo Ins i u e o Technol-
ogy. He is he au ho o Se ice Robo ics wi hin he
Digi al Home (Sp inge , 2011), A P og ama se Ap ende
Jugando (Pa anin o, 2017), Sis emas de Medida y Regu-
lación (Pa anin o, 2018), and Model P edic i e Con ol
(Sp inge , 2025). He is also he edi o o Dis ibu ed
Model P edic i e Con ol Made Easy (Sp inge , 2014)
and Con ol Sys ems Benchma ks (Sp inge , 2025). His
esea ch ocuses on he con ol o dis ibu ed cybe -physical sys ems, wi h a
special emphasis on in eg a ing he e ogeneous agen s in o he con ol loop. He
has published mo e han 200 jou nal and con e ence pape s and has led mul iple
esea ch p ojec s. His achie emen s ha e been ecognized wi h se e al awa ds
and hono s, including he Spanish Royal Academy o Enginee ing’s medal o his
con ibu ions o p edic i e con ol in la ge-scale sys ems and he dis inc ion o
becoming he younges ull p o esso in he Spanish uni e si y sys em in 2020.
9
