Ac i e Se P edic ion using Machine Lea ning Me hods
o Complexi y Reduc ion in Nonlinea MPC
Raphael Dy ska∗a, Ka ol Kišb, Ma in Klaučob, and Ma in
Mönnigmanna
aAu oma ic Con ol and Sys ems Theo y, Ruh -Uni e si ä Bochum, Ge many
bIns i u e o In o ma ion Enginee ing, Au oma ion, and Ma hema ics, Slo ak
Uni e si y o Technology in B a isla a, Slo akia
No embe 2025
P ep in
Abs ac
We use Machine Lea ning (ML) me hods o simpli y he Nonlinea P o-
g am (NLP) a ising in Nonlinea Model P edic i e Con ol (nonlinea MPC,
NMPC). Since e e y solu ion o an NMPC p oblem de ines a se o ac i e
and a se o inac i e cons ain s, we p opose o use p edic ions abou hese
se s o educe he complexi y o he NLP o be sol ed. Speci ically, we use
ML me hods o p edic ac i e se s o NMPC p oblems. The equi ed clas-
si ica ion ne wo ks a e simple enough o be e alua ed online, i.e., du ing he
un ime o he con olle . They can be ained o a high accu acy, quali ying
as sui able candida es o an applica ion o NMPC. The esul s a e e alu-
a ed using nume ical simula ions o a model o a con inuous s i ed- ank
eac o .
Keywo ds: nonlinea model p edic i e con ol, machine lea ning, neu al
ne wo ks, classi ica ion, ac i e se s
∗Co esponding au ho . Email add ess: [email p o ec ed]
1
1 In oduc ion
In au oma ic con ol, he concep o Nonlinea Model P edic i e Con ol
(Nonlinea MPC, NMPC) was one o he main esea ch opics add essed du -
ing he las decades, due o he abili y o NMPC o handle bo h cons ain s
and sys ems wi h mul iple inpu s and ou pu s (see, e.g., [1, 2, 3, 4, 5]). In
NMPC, he con ol signal applied o he sys em is de e mined by sol ing
a cons ained nonlinea p og am (NLP) in e e y ime s ep. Sol ing he
NLP in e e y ime s ep can be compu a ionally demanding and migh be
es ic i e o ce ain applica ions. Thus, in ense esea ch ocused on ap-
p oaches educing he compu a ional e o o sol ing he unde lying NLP.
In he classic NMPC li e a u e, hese app oaches can be di ided in o con-
ibu ions add essing he algo i hmic pe o mance o speed-up he solu ion
p ocess i sel (see, e.g., [6, 7]) and app oaches exploi ing in o ma ion abou
he s uc u e o he solu ion o ei he simpli y he unde lying NLP be o e
sol ing i (see, e.g., [8, 9]) o o e en a oid i s solu ion (see, e.g., [10, 11, 12]).
The combina ion o linea and nonlinea MPC wi h Machine Lea ning
(ML) me hods is ano he well-s udied ield. Many con ibu ions use neu al
ne wo ks o eplace he con olle comple ely (see, e.g., [13, 14, 15, 16, 17]).
Howe e , ML me hods also ha e he po en ial o assis NMPC and ice-
e sa. ML me hods a e, o example, o en used o o m he p edic ion
model (see, e.g., [18]) o o une he cos unc ion o he op imiza ion p ob-
lem (see, e.g., [19]). Con e sely, MPC me hods can be used o p o ide
sa e y gua an ees o lea ning-based con olle s (see, e.g., [20]). Mo e e-
cen app oaches use ML me hods o cope wi h unce ain ies, by p edic ing
plan -model misma ch, o example, bu also o he sou ces o unce ain y
(see, e.g., [21, 22, 23]).
Ano he b anch o app oaches uses ML me hods o accele a e he solu-
ion o he MPC p oblem (see, e.g., [24, 25]). In MPC, a solu ion o he
unde lying op imiza ion p oblem de e mines he ac i e se , i.e., he se o
inequali y cons ain s ha ac ually hold wi h equali y a he op imal solu-
ion. The au ho s in [26] p opose o use classi ica ion algo i hms o p edic
he se o cons ain s ha a e expec ed o be ac i e o he op imal solu ion
o a linea MPC p oblem and o use his p edic ion o wa m-s a ing an
ac i e-se algo i hm used o sol ing he unde lying op imiza ion p oblem.
In his a icle, we ex end he idea o p edic ing ac i e se s using ML
2
me hods as p esen ed by [26] om linea o nonlinea MPC p oblems. How-
e e , ins ead o using he p edic ed ac i e se s on an algo i hmic le el as
p oposed by [26], we use a s aigh o wa d app oach o simpli ying he
NLP be o e sol ing i . In p e ious wo ks (see [8, 27], see also [28, 29]) we
used a p io i in o ma ion abou ac i e and hus inac i e cons ain s o sim-
pli y he unde lying NLP by ea anging and emo ing cons ain s. While
he p edic ion o he ac i e se was based on he p inciple o op imali y o
NMPC on a sh inking ho izon in [27], we used p edic ions based on he
solu ion o he p e ious ime s ep o simpli y he NLP also o he classic,
eceding ho izon case in [8]. Following hese wo ks, we now p opose o use
ML me hods o p edic ac i e cons ain s and o apply he same simpli ica-
ion app oach o he unde lying NLP. The non-con exi y o he unde lying
NLP poses challenges o he gene a ion o aining da a since, o exam-
ple, se e al local minima wi h he same ac i e se and se e al local minima
wi h di e en ac i e se s may exis o he same ini ial s a e (see [30] o
p ope ies o NMPC solu ions esul ing om non-con exi y).
In Sec ions 2.1 and 2.2, we s a e he class o NMPC p oblems in es iga ed
he e and in oduce a model o a con inuous s i ed- ank eac o (CSTR) ha
se es as an example h oughou he pape , espec i ely. Sec ion 3 s a es he
me hods applied in his a icle, speci ically he p ocess o gene a ing aining
da a (Sec. 3.1), he ype o classi ica ion ne wo ks and hei aining o
ac i e cons ain p edic ion (Sec. 3.2), and he simpli ica ion app oach o a
known se o ac i e cons ain s (Sec. 3.3). In Sec ion 4, we p o ide closed-
loop simula ion esul s o he CSTR in oduced in Sec ion 2.2. Conclusions
and an ou look o u u e wo k a e gi en in Sec ion 5.
2 P oblem S a emen
2.1 NMPC P oblem Class
We conside he class o nonlinea dynamical sys ems in disc e e ime gi en
by di e ence equa ions
x(k+ 1) = (x(k), u(k)), k = 0,1,..., (1)
wi h sys em s a es x(k)∈Rnand inpu signals u(k)∈Rm. We assume he
nonlinea mapping :Rn×Rm→Rn o be wice con inuously di e en iable
3
and we assume (0,0) = 0.
To egula e sys em (1) o he o igin we sol e, in e e y ime s ep, he
cons ained nonlinea op imiza ion p oblem
min
X,U ∥x(N)∥2
P+
N−1
X
k=0 ∥x(k)∥2
Q+∥u(k)∥2
R(2a)
s. . x(k+ 1) = (x(k), u(k)), k ∈NN−1
0,(2b)
x(k)∈ X, k ∈NN−1
0,(2c)
u(k)∈ U, k ∈NN−1
0,(2d)
x(N)∈ T ,(2e)
x(0) = x(k),(2 )
o ho izon leng h N, wi h op imiza ion a iables X= (x(1)⊺, ..., x(N)⊺)⊺
and U= (u(0)⊺, ..., u(N−1)⊺)⊺desc ibing he p edic ed s a es and inpu s,
espec i ely. The ma ices P,Qand Rha e ob ious dimensions and a e
assumed o be posi i e de ini e. The se s Xand Ua e assumed o con ain
he o igin in hei in e io s and o be de ined by ini e numbe s o linea
inequali ies, which implies hey a e con ex polyhed a. The e minal se T
in (2e) is an ellipsoid he e (see, e.g., [31] o his p oblem class)
T={x|x⊺Px ≤γ},(3)
which implies i is ep esen ed by one addi ional inequali y. We assume
he e exis s a s abilizing con ol law κ(x)in Tsuch ha he e minal cos
unc ion ∥x(N)∥2
P, he e minal se T, and κ(x) ul ill he s abili y condi ions
summa ized as in [1], A1–A4.
To simpli y he no a ion, we in oduce z= (X⊺, U⊺)⊺and ew i e he
op imiza ion p oblem (2) in he compac o m
min
z∥z∥2
H(4a)
s. . F(x(0), z)=0,(4b)
G(x(0), z)≤0,(4c)
wi h H∈RN(n+m)×N(n+m),H≻0. In (4), he equali y cons ain s a e
4
abb e ia ed as
F(x(0), z) :=
x(1) − (x(0), u(0))
.
.
.
x(N)− (x(N−1), u(N−1))
,
and he inequali y cons ain s desc ibed by (2c)–(2e) a e summa ized in
G(x(0), z).
Le he op imal solu ion o (4) o an ini ial s a e x(0) be deno ed by
z⋆(x(0)), which we o en abb e ia e by z⋆. I no s a ed o he wise, he op i-
mal solu ion o (4) is assumed o be a local minimum. Le Q={1, ..., q}de-
no e he se o indices o all qcons ain s. A cons ain i∈ Q is called ac i e
o he op imize z⋆i ow io Gis ze o, i.e. Gi(x(0), z⋆)=0. We ecall ha
inac i e cons ain s a e ul illed wi h s ic inequali y, i.e., Gi(x(0), z⋆)<0.
We in oduce he ac i e and inac i e se Aand Icollec ing he indices o
ac i e and inac i e cons ain s, espec i ely. We w i e A(x) o an ac i e
se ha esul s o he ini ial s a e x.
2.2 Benchma k P ocess: Con inuous S i ed-Tank Reac o
We use he ollowing CSTR model [32] as an illus a ing example h oughou
he pape :
dcA
d =q
V(cA, −cA)−k0e−E
RT cA(5a)
dT
d =q
V(TA, −T)−∆Hk0
ρcp
e−E
RT cA+UA
V ρcp
(Tc−T).(5b)
S a e a iables cAand Tdesc ibe he concen a ion o subs ance Aand
he eac o empe a u e, espec i ely, and he manipula ed a iable Tcis
he empe a u e o he coolan s eam. Pa ame e alues a e collec ed in
Table 1.
The con ol ask is o egula e he sys em o an uns able s eady s a e.
Following [32], we de ine he s a e and inpu ec o s
x1=cA−0.5
0.5, x2=T−350
20 , u =Tc−300
20
by shi ing he a iables o he uns able equilib ium. The sys em is dis-
c e ized wi h a sampling ime o Ts= 0.03 min, and he weigh ing ma ices
5
Table 1: Pa ame e alues o he CSTR.
pa ame e alue pa ame e alue
ρ1000 g L−1cp0.239 J g−1K−1
∆H−5×104J mol−1E
R8750 K
k07.2×1010 min−1UA5×104J min−1K−1
q100 L min−1TA, 350 K
V100 L cA, 1.0 mol L−1
in he NMPC cos unc ion (2a) and he p edic ion ho izon a e chosen o be
P= 265.88 49.39
49.39 125.99!, Q = 10 0
0 10!, R = 1, N = 10.(6)
Fu he mo e, we in oduce inpu and e minal cons ain s
−1≤u(k)≤3.5,(7)
x(N)∈ T ,T={x∈ X | x⊺P x ≤γ}, γ = 9.3353,(8)
and he e minal con olle as κ(x) = KTx, wi h KT= (−1.6154,−3.6644),
such ha he e minal weigh ing ma ix, he e minal se , and he e minal
con olle ul ill he s abili y condi ions e e ed o in Sec ion 2.1.
3 Ac i e Se P edic ion using Classi ica ion
We i s in oduce he gene a ion o sui able aining da a, ollowed by a
desc ip ion o he aining p ocess o a classi ica ion ne wo k Φ(x) o p e-
dic ing A(x). In he las subsec ion, we s a e a simpli ica ion app oach o
he NLP (4) as one possibili y o bene i om he p edic ion o ac i e se s.
3.1 Gene a ion o T aining Da a
Fo a eliable p edic ion o an op imal ac i e se , he aining da a should
be ep esen a i e o he expec ed ini ial s a es ha occu du ing closed-
loop con ol and conside ea u es o he sys em a hand. As shown in [30],
he non-con exi y o he NLP can lead o he exis ence o se e al local
minima wi h di e en ac i e se s o he same ini ial s a e. This may lead
o inconclusi e aining da a o pa s o he s a e space such as, e.g., a eas
6
o s a es wi h mixed ac i e se s, which will in gene al impede he aining
p ocess in e ms o con e gence and esul ing accu acy.
We showed in [30] ha he esul ing minimum s ongly depends on he
ini ial guess o wa ded o he op imize . Du ing con ol, we o en apply a
wa m-s a ha esul s om ex ending he solu ion o he p e ious ime
s ep using he e minal con olle κ(x) oge he wi h he p edic ion model.
To gene a e he aining da a, we ollow he s a egy desc ibed in [26] o
he linea case and egula e a well-dis ibu ed se o ini ial s a es closed-
loop in o he e minal se . The use o he same wa m-s a he e is essen ial
o minimize he in luence o a ying local minima co up ing he aining
da a. Fu he , by gene a ing he aining da a in a closed-loop ashion, he
highes concen a ion o da a will esul o he a ea a ound he e minal
se , which is also expec ed o be he a ea con aining mos o he s a es o
which p oblem (4) needs o be sol ed when con olling he co esponding
sys em. Fo mally, we summa ize pai s o a speci ic xiand Aias a da a
uple Di o he aining. Ini ial condi ions esul ing in he same ac i e se
will be summa ized, and we call he esul ing se a class o b e i y. All
op imiza ion p oblems sol ed o gene a ing aining and es ing da a and
du ing he compu a ional s udy in Sec ion 4 we e implemen ed in Ma lab
using he nonlinea op imize mincon and he in e io -poin algo i hm.
Fo he CSTR, we gene a ed a g id o easible ini ial s a es wi h a s ep
size o ∆x1= ∆x2= 0.01. Regula ing all s a es o he g id in o he e minal
se , in o al 134 736 da a uples o s a es and co esponding op imal ac i e
se we e de e mined. Tuples Diwi h equal ac i e se s Aiwe e summa ized
in o one class, esul ing in 185 di e en classes. Figu e 1(a) shows he e-
sul ing s a e space, whe e ini ial s a es ha yield he same op imal ac i e
se and hus belong o he same class a e shown wi h he same colo . Since
he aining da a was gene a ed by con olling all s a es o he ini ial g id
in o he e minal se T, he highes concen a ion o aining da a esul s
o he a ea a ound T.
The aining p ocess can be expec ed o bene i om a balanced aining
se . Classes wi h only ew membe s can be a sign o ou lie s o nume i-
cally uns able esul s. We op o a heu is ic educ ion o he numbe o
classes while simul aneously main aining a high esolu ion o he aining
da a. Since he da a uples o each class will be spli in o a aining and a
e i ica ion se o he in e nal aining p ocess in he a io 3:1(see Sec. 3.2
7
(a) O iginal aining da a. (b) P ocessed aining da a, wi h emo ed
s a es ma ked in ed.
Figu e 1: T aining da a o he classi ica ion ne wo k. The colo o he
s a es indica es he co esponding class, omi ed s a es a e shown in ed.
below), we p opose o emo e classes wi h 3o less pai s.
The his og am in Figu e 2 isualizes he dis ibu ion o he 134 736 s a es
in o he 185 di e en classes. Ob iously, he majo i y o s a es belongs o he
class on he e y le . A he same ime, classes on he e y igh o m only a
ac ion o he o e all simula ion da a. Remo ing classes wi h 3o less da a
uples, he numbe o classes could be educed by a ound 15.68%, om 185
o 156, while only 0.04% o all da a uples we e emo ed. The s a es o he
aining se ha ha e been emo ed a e ma ked in ed in Figu e 1(b). The
Figu e 2: Numbe o da a uple pe class. Wi h 37 478 uples, he class
labeled as ’1’ is he highes ep esen ed class. No e ha classes a e so ed
o ca dinali y and do no ep esen he enume a ion depic ed in Fig. 1.
8
posi ion nea he bounda y o he easible se indica es ha (i) nume ical
issues migh be a eason o ou lie esul s and (ii) emo ing hese s a es will
no se e ely in luence he esul s since in gene al mos ini ial s a es will be
loca ed in he in e io o he easible se a he han close o i s bounda y.
3.2 Ac i e Se Classi ica ion Ne wo k
A eed- o wa d neu al ne wo k (NN) consis ing o nL ully connec ed laye s
is used as classi ica ion ne wo k Φ(x), mapping s a es x o a co esponding
ac i e se Aby classi ying hem in o a class ep esen ing one unique com-
bina ion o ac i e cons ain s. The ans o ma ion o each hidden laye l
wi h 1≤l < nL ollows
(l)=ω(l)a(l−1) +ξ(l),(9)
wi h he ec o ized inpu o each node, weigh s ω, biases ξ, and ac i a ion
unc ion a a ge ing he p e ious laye . Fo he ac i a ion unc ion o he
hidden laye s, he Rec i ied Linea Uni (ReLU) unc ion
a(l)=ReLU( (l)),wi h ReLU( ) = max(0, ),
is chosen o cap u e nonlinea beha io in he ela ion x→ A. Fo he mul i-
class classi ica ion be ween nadi e en classes, he ac i a ion unc ion o
each inpu io he ou pu laye nLis chosen as he so max unc ion
a(nL)
i=σ( (nL)
i) = exp( (nL)
i)
Pna
c= 1 exp( (nL)
c),(10)
o gene a e a p obabili y o each o he nadi e en classes. An illus a ion
o he o e all p edic ion p ocess is shown in Figu e 3.
Weigh s Ω=(ω(1), . . . , ω(nL))and biases Ξ=(ξ(1), . . . , ξ(nL)) o all nL
laye s a e he aining pa ame e s o he classi ica ion ne wo k Φ(x). They
a e op imized by sol ing he c oss-en opy loss unc ion
min
Ω,Ξ−
na
X
c= 1
yclog σ( (nL)
c),(11)
which is a s anda d choice o mul i-class classi ica ion p oblems. In (11),
ycindica es he p esence o class cas o he inpu xi∈ Di, and he so max
9
p oblem o ain classi ica ion ne wo ks wi h a p ecision o mo e han 94%.
The p edic ed ac i e se s we e used o simpli y he nonlinea p og ams sol ed
in e e y ime s ep. Wi h a CSTR model as example, we could p esen a
educ ion o he a e age numbe o i e a ions necessa y o ind a con ol
inpu o almos 10%.
Fu u e wo k will in es iga e di e en o ms o classi ica ion ne wo ks,
and in es iga e echniques such as ans e lea ning o deal wi h, e.g., chang-
ing pa ame e s o he op imiza ion p oblem.
Acknowledgmen
This pape is unded by he Alexande on Humbold Founda ion esea ch
g oup linkage coope a ion p og am and by he Eu opean Commission un-
de he g an no. 101079342 (Fos e ing Oppo uni ies Towa ds Slo ak Excel-
lence in Ad anced Con ol o Sma Indus ies). KK and MK also g a e ully
acknowledge he con ibu ion o he Scien i ic G an Agency o he Slo ak
Republic unde he g an s VEGA 1/0545/20 and he Slo ak Resea ch and
De elopmen Agency unde he p ojec APVV-21-0019.
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