Sliding mode predictive control for chemical proces with time delay

SLIDING MODE PREDICTIVE CONTROL FOR CHEMICAL PROCESS WITH TIME DELAY Wins on Ga c´ıa-Gab´ın⁄Da ine Zamb ano ⁄E. F. Camacho ⁄⁄ ⁄Uni e sidad de Los Andes, M¶e ida 5101,Venezuela Tel : +58 2742402802, Fax: +58 2742402890, {wins ong,da ine}@ula. e ⁄⁄ Uni e sidad de Se illa, Se illa 41092 Espa˜na Tel : +34 954487347, Fax: +34 954487340 edua do@ca uja.us.es Abs ac : A design o a no el model p edic i e con olle is p esen ed. The p oposed Sliding Mode P edic i e Con ol (SMPC) algo i hm combines he design echnique o Sliding-Mode Con ol (SMC) wi h Model based P edic i e Con ol (MPC). The SMPC showed a conside able obus ness imp o emen wi h espec o MPC in he p esence o ime delay, and showed an enhanced abili y o handle se poin changes in a nonlinea p ocess. I s obus ness was e alua ed using a obus ness plo , i s pe o mance was judged using a single inpu single ou pu nonlinea mixing ank p ocess wi h a iable ime delay. Copy igh c °2005 IFAC Keywo ds: P ocess con ol, P edic i e con ol, Sliding-mode con ol, Time delay, Robus ness. 1. INTRODUCTION Time delay is a common phenomenon in indus ial p ocesses. I can be p oduced by he anspo ime equi ed o a uid o ow h ough a pipe o by he measu emen and analysis lag. The p esence o a ime delay complica es signi lcan ly he analysis and he design o eedback con olle s. I gene ally has aw ul consequences in he con- ol loop pe o mance, due o he in oduc ion o an uns able beha iou , making mo e di–cul o achie e a sa is ac o y con ol. Time delay limi s he alue o p opo ional con olle gain a which he phase angle c osses -180◦and he sys em be- came uns able, hus smalle con olle gains a e equi ed as consequence sluggish esponse in a con olled a iable is ob ained (Ogunnaike and Ray, 1994). Wi h a ime delay he con ol ac ion is based on delayed, hence obsole e, p ocess in- o ma ion usually no ep esen a i e o he cu - en p ocess ou pu . Special ca e should be aken when he p ocess ime delay- ime cons an a io is bigge han one, because PID con olle can be in- adequa e, and he e o e o he s con ol s uc u es could be equi ed. The con ol p oblem o ime delay sys ems has ecei ed conside able a en ion o e he las yea s (Richa d, 2003), Model Based P edic i e Con ol (MPC) has be- come one o he mos popula con ol me hodolo- gies o bo h indus y and academia. I has been success ully implemen ed in many indus ial ap- plica ions, showing good pe o mance. The basic idea o MPC is o calcula e a sequence o u u e con ol signals in such a way ha i minimizes a mul is age cos unc ion de lned o e a p edic ion ho izon. The pe o mance index o be op imised is he expec a ion o a quad a ic unc ion mea- su ing he dis ance be ween he p edic i e sys em ou pu and a p edic i e e e ence sequence o e he ho izon, plus a quad a ic unc ion measu ing con ol e io . In o de o implemen an MPC, Copy igh (c) 2005 IFAC. All igh s ese ed 16 h T iennial Wo ld Cong ess, P ague, Czech Republic 627 a model o he plan is used o p edic he u- u e plan ou pu s. This p edic ion is based on pas and cu en alues o he inpu and he ou pu o he plan . Time delay is inhe en ly con- side ed in he p edic ion model o MPC. Thus, his con ol s a egy is adequa e o p ocess wi h ime delay. The ad an ages o apply MPC o ime delay sys ems ha e been epo ed in li e - a u e (Maciejowski, 2001; Camacho and Bo d´ons, 1999; Le ine, 1996; Soe e boek, 1991; Bi mead e al., 1990). On he o he hand, Sliding Mode Con ol (SMC) is a echnique de i ed om Va iable S uc u e Con ol (VSC) which was s udied o iginally by U kin (U kin, 1981). Fo a b oad class o sys- ems, his kind o con ol is pa icula ly appeal- ing due o i s abili y o deal wi h nonlinea i ies, ime- a iance, as well as unce ain ies and dis u - bances, in a di ec manne in he ace o mod- elling e o s. In VSC, he con ol can modi y i s s uc u e. The design p oblem consis s o selec ing he pa ame e s o each s uc u e and de lning he a eling logic. The l s s ep in SMC is o de lne a sliding su ace, S( ), along which he p ocess can slide o lnd i s desi ed lnal alue. In gen- e al, he swi ching su ace ep esen s he sys em beha iou du ing he ansien pe iod, he e o e, i mus be designed o ep esen a desi ed sys em dynamics. The s uc u e o he con ol sys em is in en ionally al e ed as i s s a e c osses he sliding su ace in acco dance wi h a p esc ibed con ol law. The second s ep is o design he con ol law in such a way ha any s a e ou side he sliding su ace is d i en o each he su ace in lni e ime and s ay on i . SMC in ol ing ime- delay sys ems has ecei ed inc easing a en ion (Gouaisbau and Richa d, 2002; Yuanqing and Yingmin, 2003; Niu Y. and X., 2004) An algo i hm based on a iable s uc u e con ol and gene alized p edic i e con ol was p oposed in (Co adini and O lando, 1997). The sliding su ace p edic ion was made only wi h pas alues o inpu and i did no conside he u u e con ol alues. A dual mode con ol scheme combining nonlinea MPC and SMC was p esen ed in (Zhou e al., 2001). MPC was used o o ce he s a e in o a e minal egion wi hin a lni e ho izon while i is ou side he e minal egion and a sliding mode a iable s uc u e con olle is used while he s a e is inside he e minal egion. The p oposed single inpu single ou pu con olle is based on he idea o a combina ion o MPC and SMC ha also has he u u e con ol mo emen s o p edic ing he sliding su ace, his esul s in mo e p ecise p edic ions and allows he p ocess o be con olled wi h ime delay; he o he pa o he con ol law is also simple and wi h ewe pa ame- e s and hey ha e a clea meaning o uning. The SMPC is a single mode con olle , he main idea is o in oduce he p edic ion o he sliding su ace in o he con ol objec i e. SMPC has been applied o non minimum phase sys ems (Ga c´ıa-Gab´ın and Camacho, 2003). I showed ha an app o- p ia e choice o he uning pa ame e s o SMPC a oids he ins abili y p oblems o MPC when i is applied o non minimum phase sys ems. When MPC is used he con olle achie es he op imal ou pu by cancelling he plan ze os, including he uns able ze os, which leads o a loss o in e nal s abili y o he eedback sys em. SMPC shows a conside able obus ness imp o emen wi h espec o MPC in he p esence o modelling unce ain ies and dis u bances, while enhanced i s abili y o handle se poin changes in a nonlinea p ocess wi h in e se esponse. This a icle is o ganized as ollows: Sec ion 2 p esen s he de elopmen o SMPC. Sec ion 3 shows he con olle obus ness. Sec ion 4 shows he applica ion SMPC o a mixing ank. This is a nonlinea p ocess wi h a iable ime delay. Finally, he conclusions a e p esen ed. 2. SLIDING MODE PREDICTIVE CONTROL Mos SISO plan s when conside ing ope a ion a ound a pa icula se -poin and a e linea iza- ion can be desc ibed by: A(z−1)y( ) = z−dB(z−1)u( ¡1) + C(z−1)ξ( ) ∆(1) Whe e: y( )is he p ocess ou pu signal, u( ) is he p ocess inpu signal, ∆ : 1 ¡z−1,d: is he delay, ξ( ) is he ze o mean whi e noise, A(z−1) and C(z−1) a e monic polynomials, and B(z−1) is a polynomial ha has he ze os o he model. This model is known as he CARIMA Model (Con olle Au o-Reg essi e In eg a ed Mo ing- A e age). This model can be used o open- loop s able, open-loop uns able, in eg a ing, o nonminimum phase sys ems I has been a gued ha o many indus ial applica ions in which dis u bances a e non-s a iona y CARIMA model is mo e app op ia e (Cla ke e al., 1987). The mos usual case C(z−1) = 1 has been used be- cause in p ac ice he colou ing polynomial a e e y di–cul o es ima e wi h su–cien accu acy (Camacho and Bo d´ons, 1999). The ollowing new p edic i e sliding su ace is p oposed o de elop he con olle : S +j| =Ps(z−1)(y( +j| )¡w( +j)) + Qs(z−1)∆uC( +j¡1¡d) (2) whe e w( ) is ajec o y e e ence, Ps(z−1) and Qs(z−1) a e polynomials o deg ee np and nq espec i ely, gi en by, 628 Ps(z−1) = ps0+ps1z−1+... +psnp z−np (3) Qs(z−1) = qs0+qs1z−1+... +qsnq z−nq (4) The swi ching su ace ep esen s he sys em be- ha iou du ing he ansien pe iod, he e o e, i mus be designed o ep esen a desi ed sys em dynamics. Small oo s o Ps(z−1) p oduce a as dynamic in he sliding su ace. A p edic i e sliding su ace is also p esen ed in (Co adini and O lando, 1997), howe e , i does no use he u u e con ol signals ∆u( ¡1 + j) o p edic he u u e sliding su ace alues S +j| , as is done he e. No ice ha using he u u e con ol de ia ions allows be e p edic ions o he u u e alues o he sliding su ace, especially o con ol p ocess wi h ime delay. The gene al aim is ha he u u e p edic i e su ace (2) on he conside ed ho izon should be ze o and a he same ime, he con ol e io ∆uCnecessa y o doing so should be penalized. The exp ession o he objec i e unc ion is gi en by, J= N2 X j=N1 [ˆ S( +j| )]2+ Nu X j=1 λ(j)[∆uC( +j¡1)]2(5) whe e ˆ S( +j| ) is an op imum j-s ep p edic ion o he sliding su ace on da a up o ime ,N1 and N2a e he minimum and maximum p edic i e ho izons, Nuis he con ol ho izon, and λ(j) a e weigh ing sequences. The objec i e o he con- olle is o compu e he u u e con ol sequence in such way ha he u u e su ace S( +j) is d i en close o ze o. The minimiza ion o he objec i e unc ion J(N1, N2, Nu) p oduces ∆uC( ),∆uC( + 1),···,∆uC( +Nu). The lnal objec i e o con ol is o ensu e ha he con olled a iable is close o i s e e ence alue w( +j) a all imes, meaning ha e( ) mus be ze o. The p oblem o acking a e e ence alue can be educed o keeping S( ) a ze o. The closed-loop dynamic is chosen wi h he pa ame e s o polynomials Ps(z−1) and Qs(z−1). Once he sliding su ace has been selec ed, a - en ion mus be u ned o designing he con ol law ha sa is les S( ) = 0. The con ol law, ∆u( ), consis s o wo addi i e pa s, ∆uC( ), and ∆uD( ). Tha is, ∆u( )=∆uC( ) + ∆uD( ) (6) The con ol signal ∆uC( ) is gi en by a Model Based P edic i e Con ol algo i hm using (5). uD( ), inco po a es a nonlinea p edic i e elemen ha includes he swi ching elemen o he con ol law. This con ol sequence is gi en by, uD( +j) = KD S( +j| ) |S( +j| )|+ρ(7) whe e KDis a gain which is he uning pa ame e esponsible o he eaching mode, and ρis a uning pa ame e used o educe he cha e ing p oblem (Zinobe , 1994). In o de o minimize (5), he j-s ep ahead ou pu p edic ion ˆ S( +j| ) o j=N1,···, N2has been compu ed based on he in o ma ion known a ime and he u u e alues o he con ol inc emen s. The ollowing Diophan ine equa ion is conside ed, 1 = Ej(z−1)˜ A(z−1) + z−jFj(z−1) (8) The polynomial Ej(z−1) and Fj(z−1) a e uniquely de lned wi h deg ees j¡1 and na espec i ely, ˜ A(z−1) = ∆A(z−1). Using he plan model and a Diophan ine equa ion he ollow p edic ion ou pu equa ion can be ob ained, ˆy( +j) = Ej(z−1)B(z−1)∆uC( +j¡1¡d) +Fj(z−1)y( ) (9) In his exp ession ˆy( +j) is a unc ion o a known signal alue a ime and also o u u e con ol inpu s which ha e no ye been compu ed . Using a second Diophan ine equa ion (10) o dis inguish pas and u u e con ol alues, Ej(z−1)B(z−1) = Gj(z−1) + z−j¡j(z−1) (10) The polynomial Gjcon ains he l s js ep e- sponse pa ame e s o he plan model. The ol- lowing exp ession o he p edic ion is ob ained, ˆy( +j) = Gj(z−1)∆uC( +j¡1¡d) +ˆy( +j| ) (11) whe e ˆy( +j| ) is he ee esponse p edic ion o ˆy( +j) assuming ha u u e con ol inc emen s a e ime ¡1 will be ze o, ˆy( +j| ) = ¡j(z−1)∆uC( ¡1¡d) +Fj(z−1)y( ) (12) Subs i u ing Ej(z−1) o (8) in o (10) , his yields B(z−1) = z−j˜ A(z−1)¡j(z−1) + z−jFj(z−1)B(z−1) +˜ A(z−1)Gj(z−1) (13) De lne he ec o ( ), composed o he ee e- sponse p edic ions, ( ) = [ˆy( + 1 + d| ),···,ˆy( +N2+d| )]T(14) he ec o o u u e con ol inc emen s, ∆uC( ) = [∆uC( ),···,∆uC( +Nu¡1)]T(15) F om p edic ion (11) he p edic ed inpu -ou pu ela ionship o he plan can be w i en as, ˆy( ) = G∆uC( ) + ( ) (16) 629 whe e ma ix Gis composed o gks ep esponse pa ame e s o he SISO plan model. G=     g00··· 0 g1g0··· 0 . . .. . ..... . . gN2−1gN2−2··· gN2−Nu      (17) The p edic ion o he sliding su ace is ob ained subs i u ing (16) in o (2), ˆ S( ) = (PsG+Qs)∆uC( ) + Ps( s( )¡w( ))(18) Whe e he ee esponse o he sliding su ace s is gi en by, s( ) = F(z−1)y( ) + ¡(z−1)∆uC( ¡1)+ Ps−1Ps⁄e( ) + Ps−1Qs⁄∆uC( ¡1) (19) wi h he ma ices de lned as, Ps=     ps00··· 0 ps1ps0··· 0 . . .. . ..... . . 0psnp ··· ps0      (20) P⁄ s=     ps1··· psnp−1psnp ps2··· psnp 0 . . .. . .. . .... 0 0 ··· 0      (21) Qs=     qs00··· 0 qs1qs0··· 0 . . .. . ..... . . 0qsnq ··· qs0      (22) Q⁄ s=     qs1··· qsnq−1qsnq qs2··· qsnq 0 . . .. . .. . .... 0 0 ··· 0      (23) The objec i e unc ion (5) can be ew i en as, J= N2 X j=N1 [PsGuC+Ps( +P⁄ s+Q⁄ suC)¡Psw +QsuC]2+ Nu X j=1 λ(j)[∆uC( +j¡1)]2(24) The quad a ic minimiza ion o (24) becomes a di ec p oblem o linea algeb a, assuming he e a e no cons ain s on he con ol signal, which leads o, ∆uC( ) = KSMP C (w( )¡ s( )) KSMP C = [(PsG+Qs)T(PsG+Qs) + λI]−1 (PsG+Qs)TPs(25) and ∆uD( ) = "KDˆ S( +d) |ˆ S( +d)|+ρ,· · · ] [KDˆ S( +Nu+d¡1) |ˆ S( +Nu+d¡1) |+ρ#(26) lnally, he con ol signal is gi en by, ∆u( ) = KSMP C (w( )¡ s( )) + ∆uD( ) (27) To summa ize, he SMPC has wo pa s. ∆uD( ), and esponsible o guiding he sys em o he slid- ing su ace, and ∆uC( ) de eloped like an MPC, which is esponsible o keeping he con olled a iable on he e e ence alue. No e ha choosing Ps(z−1) = 1, Qs(z−1) = 0, he objec i e unc ion is educed J= N2 X j=N1 [ˆy( +j| )¡w( +j)]2+ Nu X j=1 λ(j)[∆uC( +j¡1)]2(28) and he Kma ix gain is he usual linea MPC, KMP C = (GTG+λI)−1GT(29) 3. CONTROLLER ROBUSTNESS This sec ion compa ed he obus ness o he con- olle p oposed agains a MPC. This is e alua ed using a obus ness plo (Shinskey, 1990). In he obus ness plo he s abili y limi o a closed loop is plo ed on loga i hmic coo dina es when he p ocess ime delay and gain change. The axes o his plo a e he delay a io ( he p ocess ime de- lay o e he p ocess ime delay he con olle was uned) and he gain a io ( he p ocess gain o e he p ocess gain he con olle was uned). All combina ion o delay a io and gain a io wi hin a ac o o wo in ei he di ec ion a e enclosed wi hin a obus ness window. I he s abili y limi o a closed loop s ays ou side o his window, ha loop is conside ed o be obus . To plo he s abil- i y limi , a con olle is l s uned o a base se o p ocess pa ame e s, ep esen ed by he middle poin (1,1) on he coo dina es. The p ocess ime delay is hen changed in a io o he base ime delay un il uni o m oscilla ions a e p oduced; his is he delay a io a he limi o s abili y. Simila ly, he p ocess gain is changed in a io o he base gain un il he limi o s abili y can be eached. O bo h can be changed simul aneously. The locus o such poin s ep esen s he s abili y en elope o ha pa icula combina ion o p ocess, con olle and uning. The obus ness plo was cons uc ed ini ially un- ing bo h con olle s o ha e simila se ling ime 630 0.4 0.5 0.6 0.7 0.8 0.9 123 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 Gain a io Delay a io SMPC MPC Fig. 1. Robus ness plo using he ollows model, ( No e ha con ollabili y a io is one ( o τ= 1)) Gp=1 s+ 1e−1s(30) F om Figu e 1 he ollowing alues o Robus - ness Index (RI) we e ob ained (MPC = 0.78 and SMPC = 1.41). The esul s shown in his sec- ion demons a ed ha he p oposed SMPC has a mo e obus beha iou han he MPC. 4. SIMULATION EXAMPLE In o de o compa e he SMPC and MPC pe - o mances a nonlinea model o a mixing ank p ocess (Camacho and Smi h, 2000), has been conside ed. he mixing ank ecei es wo s eams: a ho s eam, W1( ), and a cold s eam W2( ) (Figu e 2). The ou le empe a u e is measu ed a a poin 125 downs eam om he ank, his p oduces a a iable ime delay. The ollowing assump ions a e aken: The liquid olume in he ank is conside ed cons an , he ank con en s a e well mixed, he ank and he pipe a e well insu- la ed. The empe a u e ansmi e is calib a ed o a ange o 100 o 200 ◦F. Ho low Cold low TT Re e ence W1( ) W2( ) T3( ) T4( ) T1( ) T2( ) TC Fig. 2. Mixing ank The SMPC and MPC we e uned ini ially o ha e simila se ling ime when pe u ba ions a ound he ope a ing poin we e simula ed. The ollowing uning pa ame e s we e ob ained: MPC: N1= 40; N2= 90; Nu= 25 y λ= 20, SMPC: N1= 40; N2= 90; Nu= 20; λ= 2; Ps= 1 ¡0.9z−1;Qs= 0.1z−1;KD=¡0.2 y ρ= 0.3, bo h p edic i e con olle s use he same p edic ion model: Gp=¡0.95 2.52s+ 1e−4s(31) In his ope a ing poin he p ocess has a con ol- labili y a io bigge han one ( o τ= 1.6). I is well known ha his p oduces con ol di–cul ies (Smi h and Co ipio, 1997). Figu e 3 shows ime delay changes when he ho s eam was modi led. The ow o ho wa e changes om 250 lb/min o 225 lb/min, hen o 200 lb/min, and lnally o 175 lb/min. Time delay was inc eased a 20 % wi h hese a ia ions o he ho s eam. 050 100 150 200 160 180 200 220 240 260 Time [min] W1 [lb/min] 050 100 150 200 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 Time [min] Delay [min] Fig. 3. Time delay changes s ho s eam a ia- ions W1( ) The esponse o he empe a u e (con olled a i- able) is shown in Figu e 4. The ou pu ime cou ses clea ly shown ha as he ope a ing con- di ions change, he pe o mance o he MPC con- olle deg aded, while he SMPC main ains i s pe o mance and s abili y. In his case, as he ow o ho wa e dec eases, wi h a co esponding dec ease in cold wa e , he ime delay be ween he ank and he empe a u e senso inc eases. This inc emen in ime delay ce ainly ad e sely a iec s he pe o mance o he MPC con olle . In he Figu e 5 he p oposed con olle is compa ed wi h a s anda d PI plus Smi h P edic o (PI+SP), bo h con olle ha e been unning ini ially wi h he same se ling ime. When ime-delay was inc eased SMPC shows be e pe o mance ha PI+SP, also SMPC can use he u u e se poin knowledge and can inco po a e cons ain s. 5. CONCLUSIONS The p oposed single inpu single ou pu SMPC algo i hm combines he design echnique o SMC 631 050 100 150 200 42 44 46 48 50 52 54 56 58 Time [min] TO [%] MPC SMPC Fig. 4. SMPC and MPC pe o mance o W1( ) changes 050 100 150 200 42 44 46 48 50 52 54 56 58 Time [min] TO [%] PI+SP SMPC Fig. 5. SMPC and PI+SP pe o mance o W1( ) changes and MPC. I is shown ha mixing bo h con ol echniques esul s in a new p edic i e con ol wi h a be e obus ness han a classic MPC. The o- bus ness index quan i a i ely demons a ed ha SMPC is mo e obus han classic MPC. The con olle pe o mance using a nonlinea p ocess wi h a iable ime delay shows a conside able imp o emen in obus ness wi h espec o MPC in he p esence o modelling unce ain ies, and dis u bances and enhanced i s abili y o han- dling se poin changes. The SMPC acqui es he main ad an ages o he wo con ol me hods, he obus ness ea u es o sliding mode con ol and he good pe o mance o model p edic i e con ol. The SMPC imp o es he closed loop beha iou o MPC a oiding he s ong con ol mo emen s o SMC. The compu a ional equi emen s o SMPC a e simila o hose needed o MPC. 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