On Stiction Compensation Methods for Practical Nonlinear MPC Implementations

Copy igh © 2018 IEEE. Pe sonal use o his ma e ial is pe mi ed. Pe mission om IEEE mus be ob ained o all o he uses, in any cu en o u u e media, including ep in ing/ epublishing his ma e ial o ad e ising o p omo ional pu poses, c ea ing new collec i e wo ks, o esale o edis ibu ion o se e s o lis s, o euse o any copy igh ed componen o his wo k in o he wo ks. This is a p ep in e sion which may con ain e a a. The published e sion is a ailable in IEEE Xplo e. DOI: 10.1109/ICSTCC.2018.8540748 *Resea ch suppo ed by he Eu opean Union and he Spanish Go e n- men wi h p ojec INOPTCON (MINECO/FEDER DPI2015-70975-P). 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. REFERENCES [1] T. Hägglund, “Au oma ic on-line es ima ion o backlash in con ol loops,” Jou nal o P ocess Con ol, ol. 17 (6), pp. 489–499, 2007. [2] M.S. Choudhu y, S.L. Shah, N. Tho nhill and D.S. Shook, “Au oma ic de ec ion and quan i ica ion o s ic ion in con ol al es,” Con ol Engi- nee ing P ac ice, ol. 14 (12), pp. 1395-1412, 2006. [3] X. Zhao and Y. Tan, “Neu al ne wo k based iden i ica ion o P eisach - ype hys e esis in piezoelec ic ac ua o using hys e e ic ope a o ,” Sen- so s and Ac ua o s A: Physical, ol. 126 (2), pp. 306-311, 2006. [4] M.C. Rod íguez Liñán, “New con ol design and analysis echniques o plan s wi h ac ua o nonlinea i ies,” Ph.D. disse a ion, School o Elec i- cal and Elec onic Eng., Uni e si y o Manches e , UK, 2013. [5] R.B. di Capaci and C. Scali, “Re iew and compa ison o echniques o analysis o al e s ic ion: F om modeling o sma diagnosis,” Chem. Enginee ing Resea ch and Design, ol. 130, pp. 230-265, 2018. [6] H. Du and, R. Pa ke , A. Alanqa and P.D. Ch is o ides, “Elucida ing and handling e ec s o al e-induced nonlinea i ies in indus ial eed- back con ol loops,” Compu e s & Chemical Eng., In P ess, 2017. [7] P. Mish a, V. Kuma and K.P.S. Rana, “An online uned no el nonlinea PI con olle o s ic ion compensa ion in pneuma ic con ol al es,” ISA T ansac ions, ol. 58, pp. 434-445, 2015. [8] R.B. di Capaci, M. Vacca i and G. Pannocchia, “A al e s ic ion ole an o mula ion o MPC o indus ial p ocesses,” IFAC-Pape sOnLine, ol. 50 (1), pp. 9044-9049, 2017. [9] M. Ale Mohammad and B. Huang, “Compensa ion o con ol al e s ic ion h ough con olle uning,” Jou nal o P ocess Con ol, ol. 22, pp. 1800-1819, 2012. [10] T. Hägglund, “A ic ion compensa o o pneuma ic con ol al es,” Jou nal o P ocess Con ol, ol. 12, pp. 897-904, 2002. [11] R. S ini asan and R. Rengaswamy, “App oaches o e icien s ic ion compensa ion in he p ocess con ol al es,” Compu e s & Chemical Enginee ing, ol. 32, pp. 218-229, 2008. [12] L. Z. X. I an and S. Lakshmina ayanan, “A new uni ied app oach o al e s ic ion quan i ica ion and compensa ion,” Indus ial & Enginee - ing Chemis y Resea ch, ol. 48, pp. 3474-3483, 2009. [13] M.A. de Souza L. Cuad os, C.J. Muna o and S. Muna e o, “No el Mod- el-F ee App oach o S ic ion Compensa ion in Con ol Val es,” Indus- ial & Eng. Chemis y Resea ch, ol. 51 (25), pp. 8465-8476, 2012. [14] N. F. Tho nhill and A. Ho ch, “Ad ances and new di ec ions in plan - wide dis u bance de ec ion and diagnosis,” Con ol Enginee ing P ac- ice, ol. 15, pp. 1196-1206, 2007. [15] Q.P. He, J. Wang, M. Po mann, and S.J. Qin, “A cu e i ing me hod o de ec ing al e s ic ion in oscilla ing con ol loops,” Indus ial & En- ginee ing Chemis y Resea ch, ol. 46, pp. 4549-4560, 2007. [16] Q.P. He and J. Wang, “Val e s ic ion modeling: Fi s -p inciples s. da a- d i e app oaches,” in P oc. o he 7 h Ame ican Con ol Con ., Bal imo e, 2010, pp. 3777-3782. [17] H. Zabi i and Y. Samyudia, “MIQP-Based MPC in he P esence o Con ol Val e S ic ion,” Chem. P od. & P ocess Model., ol. 4(3), 2009. [18] H. Du and and P.D. Ch is o ides, “Ac ua o s ic ion compensa ion ia model p edic i e con ol o nonlinea p ocesses,” AIChE Jou nal, ol. 62 (6), pp. 2004-2023, 2016. [19] H. Du and and P.D. Ch is o ides, “Empi ical modeling o con ol al e laye wi h applica ion o model p edic i e con ol-based s ic ion com- pensa ion,” IFAC-Pape sOnLine, ol. 49 (18), pp. 41-46, 2016. [20] H. Zabi i and Y. Samyudia, “A hyb id o mula ion and design o model p edic i e con ol o sys ems unde ac ua o sa u a ion and backlash,” Jou nal o P ocess Con ol, ol. 16, pp. 693-709, 2006. [21] J. No ak and P. Chalupa, “Compensa ion o al e deadzone using mixed in ege p edic i e con ol,” in P oc. o he 31s Eu opean Con e ence on Modelling and Simula ion ECMS 2017, pp. 379-383, Budapes , 2017. [22] J. Ande sson, J. Åkesson, and M. Diehl, “CasADi: A Symbolic Package o Au oma ic Di e en ia ion and Op imal Con ol,” in Recen Ad ances in Algo i hmic Di e en ia ion, Sp inge , 2012, pp. 297–307. [23] O. Melande and A. Rasmuson, “Ma hema ical modeling o hea , mass, and momen um anspo in MDF ibe d ying,” D ying Technology, ol. 29 (1), pp. 64-72, 2010. [24] C. de P ada, D. Hose, G. Gu ie ez, and J.L. Pi a ch, “De eloping g ey- box dynamic p ocess models,” IFAC-Pape sOnLine, ol. 51 (2), pp. 523–528, 2018. [25] R. S ull, “We -bulb empe a u e om ela i e humidi y and ai empe a- u e,” Jou nal o Applied Me eo ology and Clima ology, ol. 50 (11), pp. 2267-2269, 2011. [26] G. Pannocchia, M. Gabiccini and A. A oni, “O se - ee MPC ex- plained: no el ies, sub le ies, and applica ions,” IFAC-Pape sOnLine, ol. 48 (23), pp. 342-351, 2015. [27] L.T. Biegle , Nonlinea p og amming: concep s, algo i hms, and appli- ca ions o chemical p ocesses, SIAM Se ies on Op imiza ion, 2010. [28] D. Limon, A. Fe amosca, I. Al a ado, and T. Alamo, “Nonlinea MPC o acking piece-wise cons an e e ence signals,” IEEE T ansac ions on Au oma ic Con ol, DOI:10.1109/TAC.2018.2798803 In P ess, 2018.