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On S ic ion Compensa ion Me hods o P ac ical
Nonlinea MPC Implemen a ions
José Luis Pi a ch
Sys ems Enginee ing and Au oma ic
Con ol Depa men .
Uni e sidad de Valladolid.
Valladolid, Spain
jose.pi a[email p o ec ed]
Ped o San os
Sys ems Enginee ing and Au oma ic
Con ol Depa men .
Uni e sidad de Valladolid.
Valladolid, Spain
ped o.san os.ba olom[email p o ec ed]
Césa de P ada
Ins i u e o Sus ainable P ocesses (IPS)
Sys ems Enginee ing and Au oma ic Con ol
Depa men . Uni e sidad de Valladolid
Valladolid, Spain
p ada@au om.u a.es
Abs ac —S ic ion in ac ua o s is a common issue wi hin he
p ocess indus y, which may conside ably deg ade he con ol
pe o mance in p ac ice. In pa icula , he imp o ed p edic ion
capabili ies o model p edic i e con ol (MPC) anish i such
ac ua o impe ec ions a e neglec ed in ac ual implemen a ions.
In his wo k, he au ho s e iew he mo e ecen app oaches o
s ic ion compensa ion in li e a u e and p opose wo p ac ical
al e na i es ha a e mo e sui able o use in la ge-scale nonlinea
MPC p oblems. The p oposals a e illus a ed and es ed o e a
eal indus ial case s udy: he ibe humidi y con ol in a medi-
um-densi y ib eboa d d ye .
Keywo ds—s ic ion, MPC, windup, deadband, compensa o
I. LITERATURE REVIEW & MOTIVATION
Al hough ac ua o s a e he mos p e alen elemen s in a
con ol loop, hei beha io is o en neglec ed when designing
con olle s. Howe e , hey a e no pe ec , making 𝑢≠𝑢 (see
Fig. 1) in some si ua ions, which con ibu es o a signi ican
numbe o nega i e e ec s such as acking issues, sus ained
closed-loop oscilla ions, e c. Among hese p ac ical limi a ions,
sa u a ion and a e cons ain s a e always p esen in applica-
ions, being hus he ype o ac ua o nonlinea i ies mo e suc-
cess ully handled in bo h classical eedback and ad anced
MPCs. Bu he e a e also o he undesi able issues such as s ic-
ion, backlash, deadband o hys e esis [1] [2] [3], ha appea
inhe en ly o he ac ua o s na u e ( low al es, mechanical
gea s, elec omagne ic de ices, e c.) and a e mo e di icul o
accoun o in he con ol design phase [4]. In his pape we will
ocus on s a ic ic ion o s ic ion, which aduces in a deadband
beha io ha can appea along he whole ac ua o ange, and is
especially ha m ul o MPC implemen a ions in he p ocess
indus y whe e he sampling ime is in he scale o minu es.
The li e a u e on s ic ion compensa ion is pa icula ly
b oad, om e iews on me hods [5][6] o applica ions [7].
Ne e heless, s ic ion compensa ion emains an ac i e esea ch
opic, especially in MPC implemen a ions [8]. F om an ex en-
si e eading, one can conclude ha , apa om exhaus i e
main enance, he igh way o handle ac ua o s ic ion is, by a ,
he use o a local con ol block be ween he main con olle
ou pu and he ac ua o in a cascade ashion (see Fig. 1). This
local compensa o (linea o nonlinea ) usually wo ks a a high-
e equency han he main con olle , p o iding hus he neces-
sa y decoupling o design he main con olle independen ly o
ac ua o s nonlinea i ies, i.e. p esuming 𝑢=𝑢. Examples o
such kind o compensa o s a e, e.g., PID-based [6][9], knocke s
[10], 2-mo e compensa o s [11] o cons an ein o cemen [12].
Figu e 1. Usual cascade con ol o compensa e ac ua o s nonlinea i ies.
The abo e lis ed compensa o s belong o he so-called mod-
el- ee me hods which, oughly speaking, add compensa ing
signals (piecewise cons an , pulses, e c.) o he con ol ones,
he eby educing p ocess a iabili y a he p ice o inc easing
he ac ua o wea [13]. This may become an impo an limi a-
ion in p ac ice because ac ua o s can p ema u ely deg ade. One
can always choose a sui able adeo be ween ac ua o wea
and con ol pe o mance, by educing he compensa ion-signal
equency, bu his leads o addi ional issues o ace: he as-
sump ion 𝑢=𝑢 is no longe alid and he inne compensa ion
loop dynamically modi ies he o e all plan beha io om he
main con olle poin o iew. Hence, e- uning o bo h con ol-
le s is equi ed o keep pe o mance close o he ideal one, bu
no clea me hodologies ha e been epo ed o his aim excep
in a ew pa icula cases [14].
The al e na i e is using model-based me hods o ep esen
he ac ua o nonlinea i ies, which go om he nai e1 [15]
𝑢(𝑡)=𝑢(𝑡)
𝑢(𝑡−1)i Δ𝑢>  | Δ𝑢<− 
o he wise  
o he mo e complex i s -p inciples ep esen a ions [6][16].
Gi en hese models, he ea ly p oposal o he s icky ac ua o
issue is o (app oxima ely) in e he s ic ion nonlinea i y in he
compensa o block so ha 𝑢≈𝑢 a each ime ins an [4].
Howe e , hese solu ions pe o m well only when Δ𝑢 is wi hin
he domain o pe ec in e sion, o he wise he sys em ge s
s uc u al misma ch and he pe o mance deg ades [8].
The clea conclusion ha can be ex ac ed om he abo e
e iew is ha he be e compensa ion, i.e. pe o mance wi h
minimum ac ua o wea , is ob ained when he plan main con-
olle is awa e o he ac ua o nonlinea i y. In his ega d, he
cu en end is achie ing s ic ion compensa ion h ough MPC,
because o i s na u al way o handle inpu cons ain s [18].
Fu he mo e, in o de o ease he nume ical esolu ion, su o-
1 He e 𝑡 ep esen s he sampling ins an ,  he highes o ce achie ed due
o s a ic ic ion and Δ𝑢=𝑢(𝑡)−𝑢(𝑡−1).
ga e models o limi ed complexi y o he ac ua o plus i s pos-
sible local compensa ion s a egy a e added o he op imiza ion
[19]. No mally hese solu ions equi e ha he MPC op imiza-
ion p oblem is sol ed wi hin small ime g ids, in o de o
p ope ly ep esen he ac ua o s dynamics, o he wise he MPC
pe o mance may s ill deg ade. Howe e , his equi emen may
impose an addi ional compu a ional s ess ha is no eally
needed in sys ems whe e he main dynamics is slow, e.g. em-
pe a u e p ocesses whe e changing con ol inpu s wi h equen-
cies lowe han minu es makes no sense.
Including ac ua o nonlinea i ies like model (1) in he MPC
can also be ma hema ically exp essed by a se ies o i - hen
ules. This leads o mixed-in ege p og amming (MIP) op imi-
za ions which combine con inuous and disc e e decisions. I is
well-known ha MIP p oblems conside ably inc ease he com-
pu a ional complexi y wi h espec o hei smoo h e sions (i.e.
wi hou disc e e decisions) which may mean a s ong limi a ion
o eal- ime implemen a ions. Ne e heless, se e al wo ks
epo ed in he li e a u e success ully ollowed his app oach i
he plan model is linea , leading o mixed-in ege quad a ic
p oblems [20][21]. The issue wi h his app oach comes om
he main limi a ion o MPC i sel : he p edic ed “op imal” pe -
o mance de e io a es unde s ong plan -model misma ch.
Thus, he abo e implemen a ions may ail in si ua ions whe e
he sys em is no well desc ibed by he linea ized model.
The e o e, wo issues emain open o esea ch: A) o a oid
he use o MIP in eal ime and B) o conside mo e accu a e
nonlinea plan desc ip ions in he MPC. The i s one has been
pa ially add essed ecen ly in [8], whe e he au ho s eplace he
ha d discon inui ies o he ac ua o model, i.e. (1), by an ap-
p oxima e s ic ion model which uses he hype bolic angen as
smoo hing unc ion:
𝑢(𝑡)=𝜂(𝑡)⋅𝑢(𝑡−1)+(1−𝜂(𝑡))⋅𝑢(𝑡) 
𝜂(𝑡)=
 anh𝜏⋅(Δ𝑢+ )+ anh𝜏⋅( −Δ𝑢) 
He e 𝜏 is he use -de ined smoo hing pa ame e o modi y
he sha pness, hence app oxima ing he o iginal model. I has
been epo ed ha he abo e model can exac ly ep oduce he
ac ua o s ic ion o la ge enough alues o 𝜏 bu , o hese
alues, cons ain s (2)-(3) a e s ill oo s i o be e ec i ely
handled by g adien -based op imiza ion sol e s. None heless, in
he abo e ci ed e e ence he au ho s appa en ly go success ul
esul s using CasADi+IPOPT [22], bu he academic example
was a a he simple linea SISO sys em. Mo eo e , no com-
men s on he ex ension/applicabili y o he p oposed app oach
o mul i a iable MIMO and/o nonlinea sys ems we e gi en.
Indeed, we ha e es ed his app oach in ou eal case s udy
(p esen ed in nex sec ion) wi h no success due o:
 The esolu ion ime o he nonlinea op imiza ion exceed-
ed by a he a ailable ime (<4 sec. in ou case).
 The p oposed “wa m s a ” inspi ed in he 2-mo e com-
pensa ion me hod wi h compu a ion o he op imal s eady-
s a e inpu alues did no p o ide accu a e esul s in p es-
ence o dis u bances and plan -model misma ch.
 The la ge scale and nonlinea na u e o he whole p oblem
oge he wi h he s i cons ain s (2)-(3) made he op imiz-
e all o en in o clea ly subop imal local minima.
These easons mo i a ed us o de i e he p ac ical ap-
p oaches p esen ed la e in Sec ion III, which pa ially add ess
he abo e open issues and we e sui able o he nonlinea MPC
implemen a ion in ou case s udy: he medium densi y ib e-
boa d (MDF) d ye , whose model de ails and con ol objec i es
a e b ie ly ou lined in he nex sec ion. Finally, some esul s
om ex ensi e simula ion es s a e p o ided in Sec ion IV and
a conclusions sec ion close he pape , o eseeing u he s eps.
II. CASE STUDY: THE MDF DRYER
The indus ial MDF d ye , ep esen ed in Fig. 2, is o med
by: a) a mix u e chambe ha ecei es a low o ho gasses
om combus ion and a cold one om he ambien , b) a he -
mally isola ed 100m-long ube whe e he ibe comes in and he
d ying akes place, and c) a cyclone s age o sepa a e ibe s
om gasses. The ai used o d ying is p o ided h ough he
p essu e gap c ea ed by a an a ached o he beginning o he
ube d ye . The lows o ho and cold ai a e egula ed by he
opening o wo laps loca ed on he ai pa hs om he hea -
eco e y sys em (𝑎) and he ambien (𝑎). These wo laps a e
he manipula ed a iables o con ol du ing no mal ope a ion.
Figu e 2. Plan diag am wi h i s con ol a chi ec u e.
Tempe a u es o he ho gases inle and he ambien a e
measu ed, as well as he en i onmen al humidi y. Ai empe a-
u es a he beginning and a he end o he ube d ye a e also
measu ed as well as he o al ai low 𝐹, in e ed by a Pi o
ube. The d y- ibe humidi y 𝑋 is he a ge a iable o con-
ol, as i signi ican ly in luences he es o he MDF p oduc-
ion p ocess. Howe e , due o he esidence ime in he cy-
clones, i s measu emen is pe o med wi h a delay o 𝑡=25 s.
Mo eo e , he ai empe a u e a he ube inle 𝑇 is equi ed o
be wi hin limi s due o ma e ial and ope a ion cons ain s.
A. G ey-box model
De ailed i s -p inciple models o MDF d ye s in ol e pa -
ial di e en ial equa ions (PDE) [23], which a e compu a ional-
ly expensi e o sol e in online MPC op imiza ions. Mo eo e ,
in ou case he esidence ime o he ibe s inside he d ye ube
is ound o be less han he sampling ime o he da a acquisi-
ion sys em (𝑡 = 5 s.), so no in o ma ion o he d ying dy-
namics inside he ube can be i ed o expe imen al da a.
The e o e, gi en hese limi a ions, we p oposed a g ey-box
lumped-pa ame e model o con ol pu poses. The model
backbone bases on mass and ene gy balances. Fo he mix u e
chambe we ha e:
𝐹 =𝐹 +𝐹 
𝐹𝐻(𝑇,𝑊)=𝐹𝐻(𝑇,𝑊)+𝐹𝐻(𝑇,𝑊) 
Whe e 𝐹 ep esen s he humid ai lows in kg/s, 𝐻(⋅,⋅) is he
speci ic en halpy unc ion in J/kg dependen on he ai empe a-
u e 𝑇 in ºC and he speci ic humidi y 𝑊 (mass o wa e ela-
i e o he o al inle ai mass) by he o mula:
𝐻(𝑇,𝑊)=1.006𝑇+(2490+1.86𝑇)𝑊 
And o he d ye ube we ha e:
𝐹 +𝐸=𝐹 
𝑞𝑋=𝐸+𝑞−𝐸𝑋 
𝐹𝐻(𝑇,𝑊)+𝑞𝐶𝑇 +𝑇 −𝑇𝑞𝐶 =
𝑄+𝐹𝐻(𝑇,𝑊)+𝑞𝑋𝐶𝑇 
Whe e 𝐸 is he e apo a ed wa e low in kg/s, 𝑞 is he inle
ibe low in kg/s, 𝑇 and 𝑇 a e he ibe inle and ou le
empe a u es, 𝑋 and 𝑋 a e he inle and ou le speci ic wa e
con en s in he ibe espec i ely, 𝐶 and 𝐶 a e he wa e
and d y wood speci ic hea s espec i ely, and inally 𝑄 is he
hea loss o he ube compu ed by:
𝑄=Δ𝑇⋅𝐴⋅𝑈 
He e 𝑈 is he hea - ans e con ec ion coe icien unde u -
bulen low2, 𝐴 is he exchange su ace and Δ𝑇 is he empe a-
u e di e ence be ween he in e nal low (ai plus ibe ) and he
ube a e aged ones ( o a oid PDEs). The e olu ion o he ube
empe a u e 𝑇 ollows he dynamics:
𝑚𝐶 +𝑚𝐶𝑇󰇗=Δ𝑇𝐴⋅𝑈−()⋅
 
Whe e 𝑚, 𝑒, 𝑚 and 𝑒 a e he mass and wid h o he
s eel ube and mine al-wool co e ing espec i ely. 𝑇 is he
a e age empe a u e o he co e ing. Mo eo e , 𝑇 is ela ed
wi h 𝑇 by he hea loss om he ube o he ambien :
()⋅⋅
=(𝑇−𝑇)⋅𝐴⋅𝑈 
He e 𝑘 is he o e all he mal conduc ion coe icien com-
pu ed om he espec i e ube wall ma e ials, and 𝑈 is he
hea - ans e con ec ion coe icien unde lamina low. The
cyclone s age is modeled by i s -o de dynamics wi h delay:
𝜏𝑋󰇗=𝑋(𝑡−𝑡)−𝑋 
Whe e 𝑋 is he inal ibe humidi y (measu able), 𝜏 is an
expe imen ally iden i ied ime cons an and 𝑡 is he delay.
The model is comple ed wi h expe imen al pa e ns ob-
ained om inpu -ou pu da a eco ded om he ac ual plan .
Essen ially we ollowed he me hodology p oposed in [24],
whe e es ima es o e ime o some in e nal (unmeasu ed)
a iables 𝑧 a e compu ed and hen eg ession cons ain s
𝑧(𝑡,𝑢,𝑦) a e i ed o hem.
In his way, expe imen al equa ions ha e been ob ained o
he o al ai low 𝐹 and o he ela ionship be ween ho and
cold lows 𝐹,𝐹, wi h espec o laps openness.
𝐹 =𝐾󰇡1−exp󰇡−7.1𝑎+𝑎󰇢󰇢 

 =𝐾
 
2 Expe imen al o mulas o i can be ound in he ela ed li e a u e.
Whe e 𝐾 and 𝐾 a e posi i e expe imen al pa ame e s. Fi-
nally, a linea ela ionship wi h he di e ence be ween he d y
bulb and we bulb ai empe a u es a he d ye ou le has been
ound he e o p edic accu a e enough he ou le ibe humidi y:
𝑋=𝛾−𝛾⋅(𝑇 −𝑇) 
𝛾=0.9𝛾+0.13 
Whe e 𝛾 is again an expe imen al pa ame e o be ob ained
om da a and 𝑇 is he we -bulb empe a u e calcula ed wi h
𝑇 and he speci ic humidi y 𝑊 [25].
The model ge s he ollowing con ol inpu s 𝑢=[𝑎,𝑎]
and dis u bances 𝑑=[𝑞,𝑇,𝑊,𝑇,𝑋] o compu e he
ou pu s 𝑦=[𝑋,𝑇,𝐹,𝑇]. Among hem, only eal- ime
measu emen s o he inle ibe humidi y 𝑋 a e no a ailable.
Hence, 𝑋 and he model pa ame e s 𝛾,𝐾 and 𝐾 a e selec ed
o be es ima ed online by a mo ing-ho izon es ima o (MHE),
ollowing he usual o se - ee MPC implemen a ion [26] [8].
B. Nonlinea MPC se up
Based on he p e ious g ey-box model, a nonlinea MPC
se up is p oposed in acco dance wi h he con ol a chi ec u e o
Fig. 2. Fo an e icien implemen a ion, he d ye dynamics in
(10) and (12) a e disc e ized by o hogonal colloca ion using 2-
deg ee in e pola ing polynomials and 5 seconds leng h ini e
elemen s [27]. Mo eo e , o enhance he esolu ion speed, a
p ope wa m s a is p o ided o IPOPT o each execu ion in a
eceding-ho izon s a egy (see Sec ion IV o de ails).
Hence, assuming ha he sys em ou pu in s eady s a e 𝑦
uni ocally de ines an equilib ium poin (𝑥,𝑢) and ha he
linea ized sys em a each easible equilib ium is con ollable,
he ollowing nonlinea MPC p oblem is se up:
min
 𝐽≔𝑤|𝑦(𝑡)−𝑦|


 +  𝑤|Δ𝑢|


 +𝑤|𝑋 −𝑋|
s. .: (4)-(16), 160<𝑇(𝑡),𝑇 <210 
 5<𝑢(𝑡),𝑢<100, Δ𝑢≤5, 𝑢(𝑡)≡𝑢 ∀𝑡≥𝐻
𝑓(𝑥,𝑢)=0, ℎ(𝑥,𝑢)=0, 𝑦=𝐶⋅𝑥
Whe e 𝐻 deno es he p edic ion ho izon, 𝐻<𝐻 is he
con ol ho izon (so ha he con olle ou pu emains cons an
in a sui able s eady s a e 𝑢 a e 𝐻), 𝜃=[𝑢(𝑡),𝑢,𝑥] a e
decision a iables, Δ𝑢=𝑢(𝑡)−𝑢(𝑡−1), [𝑤,𝑤,𝑤] a e
he weigh ing ac o s and cons ain s 𝑓(⋅) and ℎ(⋅) and a e he
model equa ions in s eady s a e, i.e. 𝑋󰇗=0,𝑇󰇗=0.
No e ha a i icial se poin s 𝑦=[𝑋,𝑇] a e in oduced
o gua an ee he easibili y o he MPC [28]. Hence, he ibe
humidi y will e en ually each he use -de ined se poin 𝑋 i
𝑤 in he o se cos is high enough.
C. S a ic ic ion in he ai laps
The MPC op imiza ion (17) only conside s ac ua o s wi h
maximum a e cons ain s, bu expe imen al da a eco ded om
he plan e eals ha he pneuma ic mechanism o con ol he
ai laps 𝑎 and 𝑎 p esen s s ic ion. Analyzing he eco ded
da a, we can oughly ep esen his phenomenon by (1), wi h
pa ame e =0.5 (in % o he o al lap openness).
III. PRACTICAL STICTION COMPENSATION METHODS
Once he op ions ha in ol e sampling a highe equen-
cies a e disca ded due o compu a ional limi a ions in MPC and
he app oach in [8] is unsuccess ul because o he easons dis-
cussed in Sec ion I, we came up wi h he ollowing wo model-
ee “p ac ical” solu ions o handle he laps s ic ion issues.
A. Windup compensa o
The i s one is a simple modi ica ion o he in eg al- e m
compensa o bu , inspi ed in [13], he compensa o ou pu is
ese o ze o when 𝑢=𝑢 (Fig. 3). In his way, he local com-
pensa o does no modi y he MPC ou pu when s ic ion is no
ac i e in he ac ua o .
Figu e 3. MPC wi h p oposed windup compensa o .
The idea behind is ha he in eg al o he ac ua ion e o
𝑒=𝑢−𝑢 o e a ime ho izon ep esen s somehow he ac ua-
ion ene gy ha he MPC should deli e o he plan in o de o
d i e i s s a e o he op imal. The e o e, wha we p opose wi h
he windup compensa o is o igge a pulse la ge han  such
ha i s ene gy du ing 𝑇 seconds app oaches he in eg al o he
e o o e he p e ious samples whe e he ac ua o is s uck,
he eby mo ing away om he poin wi h s eady-s a e e o .
This allows he MPC o e ec i ely apply (o a leas app oach)
he op imal 𝑢 in subsequen samples. The implemen able dis-
c e e- ime o m o he p oposed compensa o in Fig. 3 is
𝑢(𝑡+1)=𝑢(𝑡)+𝐾⋅𝑢(𝑡)−𝑢(𝑡) 
and 0<𝐾≤1 is he uning pa ame e .
Mo eo e , p o iding he MPC op imiza ion wi h in o -
ma ion abou he compensa o beha io can lead o imp o ed
con ol pe o mance. Using he smoo hing unc ion (3), he
MPC p oblem (17) is ex ended wi h he ollowing cons ain s3:
𝜂=
 anh(𝑢−𝜒+ )𝜏+ anh( −𝑢+𝜒)𝜏 
𝜒=𝜂⋅𝑢+𝐾⋅(𝑢−𝜒)+(1−𝜂)⋅𝑢 
Whe e he new a iables 𝜒 a e ac ually eplacing 𝑢 in he
plan model equa ions, and 𝜒, 𝑢 a e alues om he p e ious
ins an , accomplishing wi h he upda ing ules:
𝑢=𝜒; 𝜒=(1−𝜂)⋅𝜒 
Rema k. No e ha , in con as o (3), (19) does no impose s i
cons ain s on he op imiza ion, because 𝜒 and 𝑢 a e alues
compu ed a he p e ious ins an . As payback, he abo e o mu-
la ion does no allow us o in oduce he s ic ion model in he
MPC bu jus he windup compensa o , i.e. he MPC knows
abou he exis ence o 𝑢 bu does no know ha some o i s
alues along he p edic ion ho izon won’ each he plan .
3 Exp essions (19)-(21) abuse ec o no a ion o simplici y, bu hey a e
s a ed componen wise in he MPC op imiza ion p oblem.
B. Deadband penal y
In con as o he p e ious me hod, he e we a e no includ-
ing an ex e nal compensa o , bu he aim is o ell somehow he
MPC ha con ol ou pu s wi hin |Δ𝑢|<  should no be deli -
e ed because hey won’ be applied due o ac ua o s ic ion. The
aim is he same as in [8] and so, inspi ed on i , we make use
again o he smoo hing unc ion (3) bu wi hou in ol ing any
s i cons ain . The idea is, ins ead o “s ic ly” imposing con-
s ain (2)-(3), o include i as an addi ional penal y e m in he
objec i e unc ion 𝐽 o p oblem (17), so ha selec ing con ol
ac ions wi hin he deadband is mo e expensi e. Hence, he
p oposed addi ional penal y e m 𝐽 is:
𝐽≔∑𝑤𝑢⋅𝐿
𝐻𝐶−1
𝑡=0 (𝑡) 
𝐿=𝜇⋅ anh(Δ𝑢−𝜖)𝜏+ anh( −Δ𝑢)𝜏 
Whe e [𝜏,𝜏,𝜖]∈ℝ a e design pa ame e s and 𝜇>0 is
he uning pa ame e o he penal y unc ion, whose meaning is
g aphically explained in Fig. 4.
Figu e 4. Deadband penal y.
Quali a i ely, 𝜏 and 𝜏 de ine he slopes o he “s ep” edg-
es (highe slopes app oxima e disc e e decisions bu a e dele e-
ious o g adien -based op imiza ion) and 𝜖 de ines he small
ole ance o allow Δ𝑢=0. These a e ixed a p io i by he de-
signe . Only 𝜇 is le o u he uning, de ining he heigh o
he s ep. The eason he heigh is he mo e ele an uning pa-
ame e a ec ing pe o mance is g aphically gi en in Fig. 5,
whe e he deadband penal y (22) is combined wi h he usual
inpu a e quad a ic penal y in (17) as ollows:
𝐽≔𝐽𝑝+∑𝑤𝑢|Δ𝑢|2
2

  
Figu e 5. Modi ied inpu a e penal y 𝐽.
This combina ion c ea es an addi ional penal y i he sol e
selec s o apply 𝜖2≤ Δ𝑢
2≤ s
2, so i will be usually a oided. O
cou se, his is nei he a s ic es ic ion no gua an ees global
op imali y, as 𝐽𝑢 is nonco ex. Ne e heless, (17) was al eady
noncon ex due o plan model nonlinea i ies, so no specially
ha m ul solu ions a e ound i 𝜇 is sui ably chosen.
IV. SIMULATION TESTS
Some pe o mance es s we e un o quan i y he bene i s
(pe o mance loss w. . he pe ec ac ua o ) o he p oposed
app oaches. The es s we e done agains a plan simula ion ha
uses a PDE model o he d ying sec ion [23]:

 =𝑣
+
𝑈
  
Whe e 𝑙 is he spa ial a iable along he ube, 𝑣 is a pa ame-
e which depends on he d ying egime, (𝐿,𝑑) a e he leng h
and diame e o he ibe s, 𝑈 is he hea - ans e coe icien ,
(𝑇,𝑇) a e he ai d y-bulb and we -bulb empe a u es and
Δ𝐻 is he wa e apo iza ion hea . This c ea es a s ong plan -
model misma ch, bo h s uc u al and pa ame ic.
In a i s es , induced s ep changes o 3% in he se poin
𝑋; 15ºC in he gasses empe a u e 𝑇 ; 3 T/h in he ibe
low 𝑞 (usual p oduc changeo e ), 5% in he ibe inle hu-
midi y 𝑋 and 12.5% in he ibe diame e 𝑑 a e in oduced
sequen ially e e y 300 s. Fig. 6 shows he e olu ion o he
con olled humidi y 𝑋 o e ime o he ollowing scena ios: a)
pe ec ac ua o ; b) s icky ac ua o wi h no compensa ion; c)
windup compensa o ; and d) deadband penal y.
Figu e 6. E olu ion o he ou pu ibe humidi y 𝑋.
I is shown ha he es ed dis u bances a e o ally ejec ed in
he case o con ol unde pe ec ac ua o , wi h he dynamic
esponse de ined by he MPC weigh ing ac o s ( ixed o all
scena ios). O cou se, he MPC wi hou any compensa ion
leads o di e en le els o s eady-s a e e o s. On he con a y,
he e olu ion wi h he wo p oposed compensa ion me hods
app oach he esponse ob ained unde he assump ion o pe ec
ac ua o in place, al hough a small oscilla o y beha io ( ypical
om o me linea compensa o s) has been de ec ed be ween
𝑡=1100 o 𝑡=1300 s. No e ha he p oposed p ac ical ap-
p oaches do no gua an ee asymp o ic s abili y bu jus p ac ical
one, so small oscilla o y beha io can appea when he plan
becomes less con ollable. Fo ins ance, when he gasses em-
pe a u e 𝑇 inc eases signi ican ly and he ibe diame e is
small, so he smalles achie able lap mo emen (because o
s ic ion) has a bigge impac on he inal ibe humidi y.
Table I shows a compa ison o he abo e esponses unde
se e al s anda d pe o mance measu es: he in eg al squa e
e o w. . . he se poin (ISE), he in eg al absolu e e o o he
humidi y ou side con idence bands o ±5% (IAE 5%) and ±3%
(IAE 3%) a ound he se poin , he se ling ime a 95% o he
se poin s ep (Te 95%), he a e age compu a ional ime o sol e
he MPC op imiza ion (CPU) and he ac ual numbe o laps
mo emen s du ing he whole es (𝑎 Mo , 𝑎 Mo ).
TABLE I. PERFORMANCE UNDER STEP-TYPE DISTURBANCES
Pe ec
Ac ua o
No
Comp.
Ex e nal
Windup
MPC
Windup
DB
Penal y
ISE 399.67 553.96 400.76 392.55 372.52
IAE 5% 120.43 143.87 121.2 116.76 115.87
IAE 3% 274.79 340.55 278.28 270.25 264.48
Te 95% 144 s - 147 s 154 s 145 s
CPU 0.68 s 0.34 s 0.41 s 0.53 s 1.44 s
𝒂
𝒄
Mo 103 21 43 47 59
𝒂
𝒇
Mo 103 9 26 21 33
Two se s o esul s a e p esen ed in Table I o he windup
app oach (columns 4 and 5). Values in he MPC Windup col-
umn co espond o he whole app oach in Sec ion III-A whe e-
as, o comple eness, Ex e nal Windup e e s o he MPC-
unawa e basic implemen a ion o he windup compensa o
depic ed in Fig. 3, i.e. (18). The goal o showing his di e ence
was jus o s ess ou he ela i e impo ance o making he
MPC awa e o he ex e nal compensa o .
No e ha bo h p oposed compensa ion app oaches no only
imp o e pe o mance o e he s ic ion-unawa e con olle , bu
hey e en ou pe o m he si ua ion wi h pe ec ac ua o in
place. Mo eo e , his pe o mance imp o emen o e he pe -
ec si ua ion is ob ained wi h a lowe numbe o laps mo e-
men s, which educes ac ua o wea . Though his may be a
esul o sha e luck, because he MPC could become mo e
agg essi e by a di e en choice o i s weigh ing ac o s, e-
uning he main con olle is no ou in en ion. The choice o
he uning pa ame e s o he compensa ion s a egies was
𝐾=0.8 o he windup compensa o and 𝜇=0.02 (𝜏=2000,
𝜏=100) o he deadband penal y, bo h chosen ia simple ial
and e o and isual inspec ion.
TABLE II. PERFORMANCE UNDER PROCESS NOISE
Pe ec
Ac ua o
No
Comp.
MPC
Windup
DB
Penal y
ISE 96.5 130.14 99.56 94.4
IAE 5% 5.8 5.63 7.17 1.38
IAE 3% 50.78 61.16 55.07 37.03
𝒂
𝒄
Mo 104 31 62 42
𝒂
𝒇
Mo 103 10 27 15
In a second es , addi ional andom signals o sui able e-
quencies and powe s ha e been added o he senso s as well as
o he gasses empe a u e 𝑇 and o he ibe humidi y 𝑋 be-

o e he cyclones (unmeasu ed), in o de o simula e measu e-
men noise and unexpec ed ime- a ying p ocess dis u bances
simila o he ones obse ed in he ac ual acili y. In his case,
he se poin is cons an bu he all induced s ep dis u bances
we e kep . Table II shows he pe o mance alues go o his
es and Fig. 7 depic s he d ye ime esponses. The esul s in
p esence o andom p ocess noise show ha he deadband pen-
al y app oach clea ly ou pe o ms he windup compensa ion as
well as he si ua ion wi h pe ec ac ua o , bo h in e o educ-
ion (ISE, IAE) and in ac ua o wea .
Figu e 7. E olu ion o he ou pu ibe humidi y 𝑋.
In all cases, he ollowing sequence o lap inpu s is p o-
posed as wa m s a o he nonlinea MPC op imiza ion:
𝑢(1)=𝑢(0)+𝑎⋅ i 𝑢(0)≥𝑢
∗
𝑢(0)−𝑎⋅ i 𝑢(0)<𝑢
∗
𝑢(𝑡) =𝑢(𝑡)∗ ∀𝑡>1  
Whe e 𝑎>2, 𝑢(0) a e he las applied ac ions, and no a-
ion 𝑢∗ s ands o he op imal alues o 𝑢 in he p e ious un.
V. CONCLUSION
We p oposed wo p ac ical s a egies o s ic ion compensa-
ion ha a e sui able o be combined wi h gene al nonlinea
MPC o MIMO sys ems. Bo h app oaches we e es ed in an
indus ial case wi h success ul esul s, and keeping simila
esolu ion imes o he o iginal MPC implemen a ion.
Despi e hei simplici y and clea limi a ions, he p oposed
s ic ion compensa ion app oaches a e p o en o p o ide signi i-
can imp o ed con ol pe o mance wi h educed ac ua o wea
in ou case s udy, e en in p esence o he s ong plan -model
misma ch and p ocess dis u bances. Fu he mo e, he deadband
penal y app oach seems o pe o m be e han he windup
compensa o in si ua ions wi h signi ican ime- a ying dis u b-
ances and measu emen noise. Howe e , he adequa e selec ion
o he compensa o pa ame e s as well as he compensa ion
p oblem i sel emains open o u he sys ema ic esea ch.
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