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Approach of Synthesizing Model Predictive Control and Its Applications for Rotary Kiln Calcination Process
JOURNALOFIRON ANDSTEELRESEARCH,INTERNATIONAL２ ０ １ ３,２ ０( ８):１ ４ Ｇ １ ９
Appr o a cho fSyn t he s i z i ng Mod e lPr e d i c t i v eCon t r o landI t s App l i c a t i on sf o rRo t a r l nCa l c i na t i onPr o c e s s yKi ZHANG L i１ , GAO Xi an Ｇwen２
( １． Schoo lo fMe chan i c a l,El e c t r i c a landI n f o rma t i onEng i ne e r i ng,ShandongUn i ve r s i t i ha i２６４２０９, y,We Shandong,Ch i na; ２． Schoo lo fI n f o rma t i onSc i enc eandEng i ne e r i ng,No r t he a s t e r nUn i ve r s i t y, i na) Shenyang１１０８１９,L i aon i ng,Ch Ab s t r a c t:Ana c t ua lc on t r o ldemando fr o t a r i l ni st akena sba ckg r ound． Byana l z i ngandimp r ov i ngapp r oa cho fS Ｇ yk y MPC ( syn t he s i z i ngmode lp r ed i c t i vec on t r o l),ane f f e c t i ves t r a t egywh i chapp l i e sc omp l exS ＧMPCi na c t ua li ndus t r i a l , r o c e s s i sd e s i n e d ． F i r s t l a f t e ra n a l z i n h em a i nc o m o n e n t st e c h n o l o n dc a l c i n a t i o nr e a c t i o n m e c h a n i s mi n p g y y gt p gya de t a i l,t hec a l c i n i ngbe l ts t a t e Ｇ spa c emode lo fr o t a r i l ni sbu i l tus i ngPOＧMoe sp ( s t Ｇ ou t tmu l t i va r i ab l eou t t yk pa pu pu e r r o rs t a t espa c e mode li den t i f i c a t i on)me t hod．Then,c a l c i n i ngbe l tt empe r a t u r ep r ed i c t i vec on t r o lsy s t emi sde Ｇ s i on t r o lsy s t em c omb i ne st ime Ｇ de l ayga i ns chedu l ed,ou t t Ｇ t r a ck i ng,r e cu r s i vesubspa c eadap t i veand gned．Thec pu /on o t he rme t hods,andf o rmst heo f f Ｇ l i ne Ｇ l i nep r ed i c t i vec on t r o l l e ro fr o t a r i l n．Atl a s t,MATLABi sapp l i edf o r yk s imu l a t i on,expe r imen t sr uni nc ons t an tva l uet r a ck i ngands e r vot r a ck i ngs i t ua t i on．S imu l a t i onr e su l t sshowi t se f Ｇ f e c t i vene s sandf e a s i b i l i t y． Ke r d s:s t h e s i z i ngmod e lp r e d i c t i v ec on t r o l;s c h e du l e dt r a c k i ng;l i n e a rma t r i xi n e a l i t o t a r i l n;s o f ts e n s o r yn qu y;r yk ywo
S ＧMPC ( syn t he s i z i ng mode lp r ed i c t i vecon t r o l t heo r t hgua r an t e eds t ab i l i t sbe c omeanimＧ y)wi yha r t an tb r ancho ft hecu r r en tp r ed i c t i vec on t r o l．Fo r po S ＧMPCf u l l e sa sr e f e r enc et heop t ima lc on t r o l, yus Lyapunovana l i s,i nva r i an ts e tando t he r ma t ur e ys t heo r i e s,i tha sp r omo t ednew l e apsf o rp r ed i c t i ve con t r o lt heo r t udyandha sob t a i nedp l en t eousr e Ｇ ys [ ] s e a r chr e su l t s１－３ ． I no r de rt or educ et heon Ｇ l i neMPCc ompu t a t i on , [ ] bur den t het e chn i fRe f ２ wa sapp l i edo f f Ｇ queo l i ne,sucht ha tas equenc eo fne s t ede l l i o i dsand ps co r r e spond i ngs t a t ef e edba ck ga i nsa r ec ons t ruc t ed． Ther e a l Ｇ t imef e edba ck ga i ni sp r ope r l s enon Ｇ ycho l i nef r om Re f [ ４]t oRe f [ ７]．Howeve r,c ompa r ed t ot het r ad i t i ona lp r ed i c t i vec on t r o lwi de l edi n y us , i ndus t r i a lpr a c t i c e S ＧMPCha sf ewc a s e so fsuc c e s s Ｇ f u lapp l i c a t i ons．The r ea r es eve r a lva l uab l er e su l t s, sucha smu l t i Ｇmode lRMPC ( r obus tmode lp r ed i c t i ve ) con t r o l t ot hec on t r o lo fnuc l e a rs t e am gene r a t o r ands t ab l et oac on t i nuouss t i r r edt ankr e a c t o rde Ｇ [ ８－１１] v i c e si nl abo r a t o r c e s s f u l l ．Thes t a t uso f ysuc y l a cko fpr a c t i c a lapp l i c a t i oni saga i ns tt heo r i i na li n Ｇ g
t ens i ono fS ＧMPC． Thes t a r t i ngpo i n to fs t udyi nt h i spape ri st ak Ｇ i nga c t ua lcon t r o ldemando ft i c a lcomp l exi ndus Ｇ yp t r i a lp r oc e s sa s ba ckg r ound,and r e s e a r ch above r ob l ems．Fur t he rmo r e,a f t e rana l z i ngandimpr o Ｇ p y v i ng S ＧMPC me t hod,an e f f e c t i ves t r a t egy i s de Ｇ , s i i chapp l i e scomp l exS ＧMPC me t hodi na c Ｇ gned wh t ua li ndus t r i a lp r oc e s s．
１ L imeRo t a r l nPr o c e s sDe s c r i t i on yKi p
Ac t i vel imero t a ryk i l nsys t em hasmanyt e chＧ no l og i c a lproc e s s e s．Theproc e s si sma i n l d i v i d ed y i nt o ma t e r i a lsys t em and a i rf l ow sys t em．Raw ma t e r i a lbe come s produc ta f t e r pr ehea t i ng,h i gh t empe r a tur ec a l c i na t i onandcoo l i ng．Pr ima rya i r, s e conda rya i randcokeovengascons t i tut ethea i r f l ow sys t em．Se conda ry a i rc ancoo lthef i n i shed t．Exhaus tgas e sent e r pr ehea t e rand pr e Ｇ produc hea tr aw ma t e r i a l． The who l e t e chno l og i c a l e s si sshowni nF i １． I nt h i spape r,t hepr o Ｇ proc g /d, duc t i onc apa c i t fl imek i l nunde rs t udyi s６００t yo i nt hes i z eo f４ ０ m×６０ m．
Founda t i onI t em: I t em Spons o r edbyNa t i ona lNa t ur a lSc i enc eFounda t i ono fCh i na ( ６１０３４００５) B i o r aphy: ZHANG L i( １９７９—),Ma l e,Do c t o r,Le c t ur e sh i Ｇma i l:neuzhang l i＠１６３ c om; Re c e i v e dDa t e:Ap r i l２３,２０１２ p; E g
I s sue８ App r oa cho fSyn t he s i z i ngMode lPr ed i c t i veCon t r o landI t sApp l i c a t i onsf o rRo t a r l nCa l c i na t i onPr o c e s s １５ yKi
augmen t ed obs e r va t i on ma t r i x Γf ands t a t espa c e ∧
ve c t o re s t ima t i on Xf c anbeob t a i nedt hr oughSVD de compos i t i on．
３ App r oa cho fS ＧMPC
F i １ Te chno l o i c a lp r o c e s scha r to fr o t a r i l n g g yk
２ Ca l c i na t i onPr o c e s sMod e l l i ng POＧMoe sp me t hodi sus edi nc a l c i na t i onp r oc e s s mode l l i ng．Thes t a ckve c t o rs t a t espa c eequa t i oni s scu r r en tt ime i n t r oduc edi nEqn ( １)．Suppo s ek a andf a sf u t u r et ime,t hen ( ) ( yf ＝Γfx k ＋HfUf ＋GfWf ＋Vf １) éê ùú ( ) y k ê ú ê ( ) úú ê y k＋１ ú ,Γf ＝ De f i nef u t u r e ou t t yf ＝ ê pu ê ⋮ ú ê ú êê ( ) k＋f－１ úúû ë y éê éê D ０ ０ ùúú C ùúú ê ê ê ú ê CB D ０ úú ê CA ú ê ê ú , Hf ＝ ê ú, ⋮ ⋮ ⋮ ⋱ ⋮ ú ê ú ê ê ú ê ú êê êê f－１ úú f－２ C C B CAf－３B D úúû ë A û ë A o t he rs t a ckve c t o rGf ,Wf ,Vf andf u t u r ei npu tUf havet hes imi l a rpe r f o rmanc e．Al s o,pa s tt imes t a ck ve c t o rc anbede s c r i beda s: ( yp ＝Γpx( k－p)＋HpUp ＋GpWp ＋Vp ２) [ ) , ( ) ( whe r e pa s tou t typ ＝ y k－p y k－p ＋１ pu T k－１)] ando t he rve c t o r sandma t r i c e shaves imＧ y( i l a rde f i n i t i ons．Eqn ( １)andEqn ( ２)a r ewr i t t eni n : Hanke lma t r i xf o rm ( Yf ＝ΓfXf ＋HfUf ＋GfWf ＋Vf ３) ( Yp ＝ΓpXp ＋HpUp ＋GpWp ＋Vp ４) I npu tandou t tHanke lma t r i xexp r e s s eda s: pu éê y( k) y( k＋１) y( N －f＋１)ùúú ê ê ú, Yf ＝ ê ⋮ ⋮ ⋱ ⋮ ú ê ú êë ( k＋f－１) y( k＋f) y( N ) úû y whe r e Hanke l ma t r i xYp ,Uf ,Wf ,Vf ,Up ,Wp , andVp haves imi l a rde f i n i t i on．Tos o l ves t a t espa c e , mode l QR de c ompo s i t i oni s madet o Hanke l ma Ｇ t r i x． é ù é ù éê T ùú ê fú ê １１ ０ ０ ０ ú Q U L １ ú ê ú ê ú ê ê ú ê ú ê Tú Y L L２２ ０ ０ ú Q ê ２ú ê p ú ê ２１ ( ú ú ê êê úú ＝ êê ５) ê Tú U L L３２ L３３ ０ úú Q ê ３ú ê pú ê ３１ ú ê ú ê ú ê êë ú êë Y L L４２ L４３ L４４ úû êëQ４T úû ４１ f û é Tù ∧ ê ２ú Q [ ４２ L４３ ] êê T úú ＝ΓfXf ,t WhenN →∞ ,t hen L he êë úû Q３
Thef o l l owi ngs t ab i l i z ab l et ime Ｇ de l ayl i ne a rs Ｇ ys t em wi t hapo l t op i cunc e r t a i n t s c r i t i oni scon Ｇ y yde p s i de r ed: p
x( k＋１)＝A( k) x( k)＋ ΣAj ( k) x( k－ j＝１
( τj )＋B( k) u( k－τ) ６) nu nx , , r es t a t er e s e c Ｇ whe r e u∈R a r ei npu t andx∈R a p t i v e l τandτi , i∈{ １,２,,p}a r ei n t e e ri npu tand y; g |A１ ( ||Ap ( |B ( s t a t ed e l a s;[ A( k) k) k) k)]∈Ω y whe r eΩ i st heconvexs e t．As sumet ha tAj ( k),j∈ { , ,, } １ ２ r ea s t o t i c a l l t ab l eandf o rs imＧ ymp ys p a l i c i t p y,０≤τ＜τ１ ＜ ＜τp ． Thepur ei st ode s i r ed i c t i vecon t r o l l e r pos gnap , t ha tbr i ngst hes t emt ot heequ i l i br i um po i n t a t ys ch i ev i ngt hef o l l owi ngcos ti ndex: e a cht imek a mi n J∞ ( k)＝ max u( k) [ A( k＋i) k＋i) k＋i) k＋i)]∈Ω ．∀i≥０ |A１( ||Ap ( |B( ∞ ２ W i＝０
Σ [‖x( k＋τ＋i| k)‖ ＋‖u( k＋i| k)‖２R ]
k＋i| k)≤u,－ψ≤ψx( k＋τ＋ s t －u≤u( －
－
( i＋１| k)≤ψ,∀i≥０ ７) /upp whe r e,uandui si npu tc on s t r a i n t sl owe r e rbound; －
/uppe －ψandψi ss t a t econs t r a i n t sl owe r rbound;W －
andR i spe r f o rmanc ei ndexwe i tma t r i c e s． gh
３ １ Of f Ｇ l i nepr e d i c t i v ec on t r o l Toso l veEqn ( ７),t hep r ed i c t i ono fx( k＋i| k) i sne eded． I ti spos s i b l et ode f i nes e t si n wh i cht he f u t ur es t a t e swi l ll i e．De f i net hes e tχ( k＋i|k)＝Co { xli－１ l１l０ ( k＋i|k),１≤lj ≤L ,０≤j≤i－１}．As Ｇ sumi ngt ha tx( k＋i| k)∈χ( k＋i|k),i fxli－１ l１l０ ( k ＋i| k)s a t i s f i e s li－１l１l０ ( x k＋i＋１| k)＝Alixli－１ l１l０ ( k＋i| k)＋ p
li li－τj －１ ΣAj x j＝１ li li－１l１l０
l１l０
( k＋i－τj| k)＋
( ( B u k＋i－τ| k) ８) , ( ) whe r e χ k＋i|k i st het i t e s ts e tt ha tcon t a i ns gh a l lpos s i b l ef u t ur es t a t e sx( k＋i＋１|k)．Nows e l e c t Lyapunovf unc t i onr e f e r r i ngt ot he me t hodo f Ko Ｇ ha r ei n１９９６．Thes t ab i l i t t r a i n t sc anbego t: ycons T Tù é êΛ Ml Ξ ú ê ú ( êêMl Λ ０ úú ≥０, l∈ { １, ２,, L} ９) ê ú ê Ξ ０ γI úû ë , { whe r e Λ ＝d i ag Q ,Q０ ,Q１ ,,Qp },Ml ＝
J ou r na lo fI r onandS t e e lRe s e a r ch,I n t e r na t i ona l Vo l ２０
１６
l l l l éê l A Q B Y A１Q１ Ap－１Qp－１ ApQp ùúú ê ê ú ０ ０ ０ ０ Q ê ú ê ú ê ú, ０ ０ ０ ０ Ξ＝ ０ Q ê ú ê ú ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ê ú ê ú ê ú Qp－１ ０ ０ ０ ０ ë û １/２ éê ù ０ ０ú W Q ０ ０ ê ú ．I tf o l l owst ha t / êê ０ R１ ２Y ０ ０ ０úúû ë J∞ ( k)≤V [ w( k＋τ|k)]．De f i neV [ w( k＋i|k)]≤ , , γ whe r eγi sauppe rbounds c a l a r andEqn ( ９) i sequ i va l en tt o:
１ ∗ ùúú ≥０,∀l０ , l１ ,, wlτ－１ l１l０ ( k＋τ| k) Λ úúû L} ( lτ－１ ∈ { １, ２,, １０) Suppo s et he r ea r eaγ,s Z, mm e t r i c m a t r i c e s y , { , ,, , , , } Γ Q Q０ Qj j∈ １ ２ t r i xY ＝F p andma ( ha tEqn ( ９),Eqn ( １０)andt hef o l l owＧ k) Q０sucht
éê ê êê ë
i ngLMIa r es a t i s f i ed． é ù êZ ú Y ú ２ , j ≤uj êêê T １,, nu } ( １１) i ． n f, j＝ { ú ≥０ Zj Y ë Q０ úû êé Γ ∗ ∗ ∗ úù ê ú T ê ú AlQ) Q ê( ú ψ ê ú l T ú ≥０ ) ( B Y １２) êê( Q０ ψ ú ê ú ê ⋱ ú ê ú T ê( l ] Qp úû ApQp ) ë ψ whe r e,uj,inf＝ mi uj ,uj };ψs,inf ＝ mi n{ n{ s, ψ ψs}; － －
Zjji st hejt hd i agona le l emen t so fZ．Thent hecon Ｇ s t r a i n ti ss a t i s f i ed．Ateacht imeapp l i ngcont ro l y ( ) ( ) ( ) , u k ＝F k x k t hef o l l owi ngr e su l tc anbeob Ｇ t a i ned:
mi n
γ
γ, Q, Q０, Q１, , Qp , Y, Z, Γ
s t Eqn ( ９), Eqn ( １０), Eqn ( １１), Eqn ( １２) ( １３) Re f e r r i ngt ot heo f f Ｇ l i neRMPCi n Eqn ( １１),
t heo f f Ｇ l i neme t hodc anbeob t a i nede a s i l st hea l Ｇ ya r i t hm１． go １)Of f Ｇ l i n e．S e l e c ta ugme n t e ds t a t e swi ,i∈ { １, lτ－１l１l０ ,N },t ( h e ns ub s t i t u t ee a c hs t a t ew k＋τ|k) i nEqn ( １０),c ompu t et hec o r r e spond i ng { γi ,Qi , nx ( τp ＋１) T , } , { Λi Yi e l l i o i dsεi ＝ w ∈R |w Λi－１w ≤ ps －１ １}andf e edba ckga i nsFi ＝YiQi ,whe r ewi shou l d , { ,, } , s a t i s f ε ⊂ ε ∀ ∈ ２ N a n d y j j－１ j l T －１ l Λi－１ － ( Ti l∈ { １,, L} ( １４) ＋１ )Λi Ti＋１ ＞０,
２)On Ｇ l i ne．Ate a cht ime,kt ake st hef o l l owi ng s t a t ef e edba ckl aw: ïìF[ αi( k)] x( k), w( k)∈εi , ï ï ( u( k)＝F( k) x( k)＝ íï w ( k)∉εi＋１ , i≠N １５) ï ïF x( ) , ( ) k w k ∈ ε î N N whe r e,F [ αi( k)]＝αi( k) Fi ＋ [ １－αi ( k)] Fi＋１ ,and T －１ －１ w( k) { αi( k) Λi ＋ [ １－αi( k)] Λi＋１} w( k)＝１．
３ ２ Ou t t Ｇ t r a ck i ngde s i pu gn ３ ２ １ MPCf o rt rac k i ng Suppos et ha ts t emou t ty( k)i sr equ i r edt o ys pu , ( t r a ckt heg i v e nt a r e t ． A ts t e a d s t a t e z ＝ x g yr y s s, ) , ( ) us i saso l u t i ono fEqn １６ ． ( Ezs ＝Fyr １６)
éê ù ０ A０ －Inx B０ ùúú nx , ny ú ê ú, , F ＝ nd [ A０ , êê úú a C０ D０ úúû I ë ny û B０]i sthenomi na lmode l．Then,theso l ut i ono f th i sequa t i onc anbepa r ame t e r i z eda szs ＝ Mθθ and r eθ∈Rnθ i sapa r ame t e rve c t o r,and yr ＝Nθθ,whe －１ T ìï[ i fr＜nx ＋nu ï VΣ U FG V ⊥ ] Mθ ＝ íïï －１ T and V i fr＝nx ＋nu î Σ U FG ïì[ i fr＜nx ＋nu ny , nx ＋nu －r ] ï G ０ ．He r e,E ＝ Nθ ＝ íïï i fr＝nx ＋nu îG ( ) UΣVT ,r＝r ank( E ),U ∈R nx ＋ny ×r ,Σ∈Rr×r ,V ∈ ìï I i fr＝nx ＋ny ï ny ( ) ．Ma t r i x R nx ＋ny ×r and G ＝ íïï T ) ( fr＜nx ＋ny î F U ⊥ ⊥i Gi saf u l lco l umnr ankandrg ≤ny ． ３ ２ ２ De s ignofo Ｇ l i ner o bu s ts t a t eo b s e rve r ff A s imp l eLuenbe r robs e r ve rt oe s t ima t et he ge s t a t eo fEqn ( ６)i sus ed:
whe r e,E ＝
éê ê êê ë
∧
∧
x( k＋１)＝A０x( k)＋B０u( k)＋ ∧
Lp [ k)－y( k)], y( ∧
∧
( １７)
k)＝Cx( k)＋Du( k) y( ∧ nx k)∈R i st hecur r en tobs e r ve rs t a t e,and whe r ex( Lp ∈Rnx ×ny i st heobs e r ve rga i nt obede t e rmi ned．
３ ３ She du l e d Ｇ t r a ck i ng MPCde s i Al r i t hm２) gn ( go Eq n ( １)g i v e st h ee i l i b r i ums u r f a c ea ndc ompu t e s qu s ( ) t h eexpe c t edequ i l i br i um po i n tf r omyr byEqn １６ ． １)Of f Ｇ l i ne．S t a r tf r om j＝０,spe c i f st he yj a () () x j ,u j ),so l vea l Ｇ co r r e spond i ngcon t r o l l e ro f( go ( j), ( j), ( j), ( j)} , { , ,, r i t hm１ andge t γi Qi Λi Yi i＝１ () () () N ．Andt hes t a t ef e edba ckga i nFij ＝Yij ( Qij )－１i s t r o l l e ri nl ook Ｇupt ab l e．Al ongt hed i Ｇ s t o r eda sjcon ( ( s) s) ( , ) , r e c t i ono fx u choos et henex tequ i l i br i um ( ) ( ) i n t( x j＋１ ,u j＋１ )．Le tj ＝j ＋１,r epe a tt h i s po () r oc e s sun t i l ∪jM＝０ ∪iN＝０εij ,and M ＝ maxj cove r p ( ( s) s) ( , ) t heexpe c t edpo i n tx u ． ２)Gi v i ngani n i t i a lx( ０)．Le tt hes t a t ebex( k) ( ) r f o rm al i nes e a r chove r{ γij , a tt imek．Then pe ( j), ( j), ( j)} Qi Λi Yi i nt hel ook Ｇupt ab l et of i ndt hel a r Ｇ s tcon t r o l l e rj andt hel a r s tsubs c r i ti such ge ge p () () t ha tx( k)∈εij ． Imp l emen tu( k)＝Fij ( k)[ x( k)－ ( ( j)] j) x ＋u ons t em． ys
４ Ro t a r l nCa l c i n i z i ngZon eTemp e r a t u r e y Ki MPCCon t r o ls s t em y I nF i ２,t her o t a r i l ni st ake na st her e s e a r c h g yk
I s sue８ App r oa cho fSyn t he s i z i ngMode lPr ed i c t i veCon t r o landI t sApp l i c a t i onsf o rRo t a r l nCa l c i na t i onPr o c e s s １７ yKi
ob e c t,andc a l c i n i ngbe l tt empe r a t u r ep r ed i c t i vecon Ｇ j t r o ls t emi sde s i on t r o ls t emc ons i s t s ys gned．Thec ys o fs e tva l uemodu l e,c a l c i n i ngbe l tt empe r a t ur eso f t s ens o r,syn t he s i z i ng mode lp r ed i c t i vec on t r o lmod Ｇ
u l e,e t c．Asshowni nF i ３,on Ｇ l i necon t r o ls t em g ys , cons i s t so ft a r tc a l cu l a t i ngun i t op t imi z a t i onso l u Ｇ ge t i onun i t,s t a t eobs e r ve run i tandr e cur s i vesubspa c e adap t i veun i t．
F i ２ S t r u c t u r ed i a r amo fr o t a r i l nc a l c i n i ngb e l tt emp e r a t u r ec on t r o ls s t em g g yk y
F i ３ On Ｇ l i n ec on t r o ls s t emd i a r am g y g
５ S imu l a t i onExp e r imen t
t ot hes t em ma t r i xchanger angeandchangel aw ys i nt heope r a t i onp r oc e s s,i n t r oduc et hepe r t urba t i on , , , t e rmμ１ μ２ andμ３ ． So t hes t a t espa c emode lpa Ｇ r ame t e r sa r ea sf o l l ows:
I no r d e rt oc h a r a c t e r i z et h eun c e r t a i n t c c o r d i ng y,a éê ùú ( ) ０ ９ ９ ３ ６ k ０ ０ ０ １ ５ －０ ００６２ －０ ０００７ －０ ０００１ １ μ ê ú ê ú ０ ００６３ ０ ９９０６μ２( k) ０ ０１７０ －０ ００１７ －０ ０００１ ê ú ê ú ú A( k)＝ êê －０ ０００９ ０ ００２６ ０ ９７９３ －０ ０１４７ ０ ０２０９ ú ê ú ０ ００００ －０ ０００１ ０ ００１８ ０ ９９８５ －０ ００６９ ê ú ê ú ê －０ k)úû ００００ ０ ０００１ －０ ００１５ ０ ００６２ ０ ９８４８μ３( ë whe r e,μ１ ∈ [ ０ ８０,１ ２０],μ２ ∈ [ ０ ８８,１ ０７],μ３ ∈ Ts＝１０s,de l aypa r ame t e rd１ ＝３,d２ ＝１０．ConＧ [ , ] , ( ) , ( ) , ( ) [ , ] , [ ０ ９９ １ ０１ B k ＝B０ C k ＝C０ D k ＝D０ ． s t r a i n t sψ＝ ０ ０ ψ＝ １５００,１３００],u＝ [ ０,０],
I ti ssuppo s edt ha tt heunc e r t a i npa r ame t e r sa r es e Ｇ l e c t edr andoml na l l owab l er ange．Samp l i ngt ime yi
－
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u＝ [ １２０００,５００００],we i t i ng ma t r i xW ＝d i ag( １, gh
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J ou r na lo fI r onandS t e e lRe s e a r ch,I n t e r na t i ona l Vo l ２０
１,１,１,１),R ＝d i ag( ０ １,０ １)． ) １ Cons t an tva l uet r a ck i ngexpe r imen t S imu l a t i onha sob t a i nedt heou t tcu r ve so f pu t he c on t r o l l ed va r i ab l e si nc l ud i ng c a l c i n i ng z one t empe r a t u r ey１ andk i l nt a i lt empe r a t u r ey２ sucha s F i ４ ( a)and ( b)．Compa r ed wi t ht he me t hodi n g Re f [ ８],t heme t hodo ft h i spape rha sobv i ousad Ｇ van t age ssucha ssho r tr i s et imeandsma l lf l uc t ua Ｇ
t i oni ns t e adys t a t e．I nf i xedva l uet r a ck i ngs i t ua Ｇ , t i on t hecompa r i soncur ve so fga sf l owands e cond Ｇ a r i ra r eshowni nF i ４ ( c)and ( d)．Ea chs t a t e ya g componen t compa r i son cur ve si n cons t an t va l ue t r a ck i ngs i t ua t i ona r ede s c r i bedi nF i ５．Fo ru s i ng g t he ga i ns chedu l ed pr ed i c t i vecon t r o l,t hecon t r o l va r i ab l echangei nl adde rshapeexh i b i t st het r ans a c Ｇ t i onp r oc e s sa te a chi n t e rmed i a t es t e adys t a t e．
( a)Ca l c i n i ngz onet empe r a t ur eou t t; ( b)Ki l nt a i lt empe r a t ur eou t t; ( c)Coa lga sf l ow; ( d)Se c onda r i rf l ow． pu pu ya
F i ４ Compa r i s onc u r v eo fc on s t an tv a l u et r a ck i ngs i t ua t i on g
F i ５ Compa r i s onc u r v eo fs t a t ec ompon en t si nc on s t an tv a l u et r a ck i ngs i t ua t i on g
２)Se r vot r a ck i ngexpe r imen t Se tt hei n i t i a lc a l c i n i ngz onet empe r a t ur eand k i l nt a i lt empe r a t u r ea r e１０００and９００ ℃． S t udyt he ab i l i t fy１ and y２ t r a ck i ng va r i ab l eg i ven va l ue． yo Thec on t r o l l ed va r i ab l e sy１ and y２ s e r vot r a ck i ng cur vei sshowni nF i ６．Ande a chs t a t ecomponent g
compa r i soncur ve sa r eshowni nF i ７． g Thes imu l a t i onr e su l t sshowt ha tf o rt het ime va r i ngk i l ncon t r o lob e c tt hecon t r o ls t r a t egyp r o Ｇ y j edi nt h i spape rc anno ton l t r o lt empe r a t ur e pos ycon t oa ch i evet hede s i r edt a r tva l ue,bu ta l soha st he ge r obus tandf a s tt r a ck i ngab i l i t r edwi t ht he y．Compa
I s sue８ App r oa cho fSyn t he s i z i ngMode lPr ed i c t i veCon t r o landI t sApp l i c a t i onsf o rRo t a r l nCa l c i na t i onPr o c e s s １９ yKi
( a)Ca l c i n i ngz onet empe r a t ur eou t t; ( b)Ki l nt a i lt empe r a t ur eou t t． pu pu
F i ６ S e r v ot r a ck i ngs i t ua t i onc ompa r i s onc u r v e g
F i ７ S e r v ot r a ck i ngs i t ua t i onc ompa r i s onc u r v eo fs t a t ec ompon en t s g
me t hodi nRe f [ ８],t hep r e s en t me t hodha sbe t t e r r f o rmanc e． pe
６ Conc l u s i on
Th i spape ri n t r oduc e sandpu t sf o rwa r das e r i e s o fnew a l r i t hmst oana l z eandimp r ovet hep r e Ｇ go y d i c t i vec on t r o l．Andane f f e c t i ves t r a t egyi sde s i gned t oapp l omp l ex p r ed i c t i vec on t r o lt op r a c t i c a li n Ｇ yc , dus t r i a lpr o c e s s．Howeve rt he r ei ss t i l lag r e a td i s Ｇ t anc ebe twe ent her e su l t sandp r a c t i c a li ndus t r i a lap Ｇ l i c a t i ons;t h i si s ma i n l c aus et hep r ob l emt ha t p ybe t hei ndus t r i a lf i e l di smo r ec omp l ext hans imu l a t i on env i r onmen t．Howt oex t r a c tp r ob l emsf r omi ndus Ｇ t r i a lpr a c t i c eandgu i det hed i r e c t i ono ft h e o r e t i c a lr e Ｇ s e a r c hh a sb e c omet h es t udyf o c u s． Re f e r enc e s: [ １] MayneD Q,Rawl i ngJB,RaoCV． Cons t r a i nedMode lPr ed i c Ｇ t i veCon t r o l:S t ab i l i t t ima l i t J]．Au t oma t i c a,２０００, yandOp y[ ３６( ６):７８９． [ ２] Ko t ha r eM V,Ba l akr i shnanV,Mo r a r iM．Robus tCons t r a i ned Mode lPr ed i c t i veCon t r o lUs i ngL i ne a rMa t r i xI nequa l i t i e s[ J]． Au t oma t i c a,１９９６,３２( １０):１３６１．
[ ３] Cuz z o l aFC,Ge r ome lJC,Mo r a r iM．AnImp r ovedApp r oa ch f o rCons t r a i nedRobus tMode lPr ed i c t i veCon t r o l[ J]．Au t oma t Ｇ i c a,２００２,３８( ７):１１８３． [ ４] WanZY,Ko t ha r eM V．AnEf f i c i en tOf f ＧL i neFo rmu l a t i ono f
Robus tMode lPr ed i c t i veCon t r o lUs i ngL i ne a rMa t r i xI nequa l i Ｇ t i e s[ J]．Au t oma t i c a,２００３,３９( ５):８３７． [ ５] Cychowsk iM T,MahonyO． Fe edba ckMi n ＧMaxMode lPr ed i c Ｇ t i ve Con t r o l Us i ng Robus t One ＧS t ep Se t s[ J]．I n t e r na t i ona l J our na lo fSys t emsSc i enc e,２０１０,４１( ７):８１３． [ ６] Kouva r i t ak i sB,Ro s s i t e rJA,SchuurmansJ．Ef f i c i en tRobus t Pr ed i c t i veCon t r o l[ J]． IEEE Tr ans a c t i onson Au t oma t i cCon Ｇ t r o l,２０００,４５( ８):１５４５． [ ７] Bao c angD,YugengX,Ma r c i nT,e ta l．ASyn t he s i sApp r oa ch
f o rOu t tFe edba ckRobus tCons t r a i nedMode lPr ed i c t i veCon Ｇ pu t r o l[ J]．Au t oma t i c a,２００８,４４( １):２５８． [ ８] WANZhao Ｇ e shV Ko t ha r e．AFr amewo rkf o rDe Ｇ yang,Mayur s i fSchedu l ed Ou t tFe edba ck Mode lPr ed i c t i ve Con t r o l gno pu [ J]． J our na lo fPr o c e s sCon t r o l,２００８,１８( ３/４):３９１． [ ９] KE Hu,YUANJ i ng Ｇ i．Mu l t i ＧMode lPr ed i c t i veCon t r o lMe t h Ｇ q odf o rNuc l e a rS t e am Gene r a t o rWa t e rLeve l[ J]．Ene r Ｇ gyCon ve r s i onand Managemen t,２００８,４９( ５):１１６７． [ １０] Gr ube rJK,Rami r e zD R,Al amo T,e ta l．Mi n ＧMax MPC
Ba s edonanUppe rBoundo ft heWo r s tCa s eCo s tWi t hGua r Ｇ an t e edS t ab i l i t l i c a t i ont oaP i l o tP l an t[ J]．J our na lo f y．App Pr o c e s sCon t r o l,２０１１,２１( １):１９４． [ １１] Di ngB,HuangB．Cons t r a i nedRobus tMode lPr ed i c t i veCon Ｇ t r o lf o rTime ＧDe l aySys t ems Wi t hPo l t op i cDe s c r i t i on [ J]． y p I n t e r na t i ona lJ our na lo fCon t r o l,２００７,８０( ４):５０９．
JOURNALOFIRON ANDSTEELRESEARCH,INTERNATIONAL２ ０ １ ３,２ ０( ８):１ ４ Ｇ １ ９
Appr o a cho fSyn t he s i z i ng Mod e lPr e d i c t i v eCon t r o landI t s App l i c a t i on sf o rRo t a r l nCa l c i na t i onPr o c e s s yKi ZHANG L i１ , GAO Xi an Ｇwen２
( １． Schoo lo fMe chan i c a l,El e c t r i c a landI n f o rma t i onEng i ne e r i ng,ShandongUn i ve r s i t i ha i２６４２０９, y,We Shandong,Ch i na; ２． Schoo lo fI n f o rma t i onSc i enc eandEng i ne e r i ng,No r t he a s t e r nUn i ve r s i t y, i na) Shenyang１１０８１９,L i aon i ng,Ch Ab s t r a c t:Ana c t ua lc on t r o ldemando fr o t a r i l ni st akena sba ckg r ound． Byana l z i ngandimp r ov i ngapp r oa cho fS Ｇ yk y MPC ( syn t he s i z i ngmode lp r ed i c t i vec on t r o l),ane f f e c t i ves t r a t egywh i chapp l i e sc omp l exS ＧMPCi na c t ua li ndus t r i a l , r o c e s s i sd e s i n e d ． F i r s t l a f t e ra n a l z i n h em a i nc o m o n e n t st e c h n o l o n dc a l c i n a t i o nr e a c t i o n m e c h a n i s mi n p g y y gt p gya de t a i l,t hec a l c i n i ngbe l ts t a t e Ｇ spa c emode lo fr o t a r i l ni sbu i l tus i ngPOＧMoe sp ( s t Ｇ ou t tmu l t i va r i ab l eou t t yk pa pu pu e r r o rs t a t espa c e mode li den t i f i c a t i on)me t hod．Then,c a l c i n i ngbe l tt empe r a t u r ep r ed i c t i vec on t r o lsy s t emi sde Ｇ s i on t r o lsy s t em c omb i ne st ime Ｇ de l ayga i ns chedu l ed,ou t t Ｇ t r a ck i ng,r e cu r s i vesubspa c eadap t i veand gned．Thec pu /on o t he rme t hods,andf o rmst heo f f Ｇ l i ne Ｇ l i nep r ed i c t i vec on t r o l l e ro fr o t a r i l n．Atl a s t,MATLABi sapp l i edf o r yk s imu l a t i on,expe r imen t sr uni nc ons t an tva l uet r a ck i ngands e r vot r a ck i ngs i t ua t i on．S imu l a t i onr e su l t sshowi t se f Ｇ f e c t i vene s sandf e a s i b i l i t y． Ke r d s:s t h e s i z i ngmod e lp r e d i c t i v ec on t r o l;s c h e du l e dt r a c k i ng;l i n e a rma t r i xi n e a l i t o t a r i l n;s o f ts e n s o r yn qu y;r yk ywo
S ＧMPC ( syn t he s i z i ng mode lp r ed i c t i vecon t r o l t heo r t hgua r an t e eds t ab i l i t sbe c omeanimＧ y)wi yha r t an tb r ancho ft hecu r r en tp r ed i c t i vec on t r o l．Fo r po S ＧMPCf u l l e sa sr e f e r enc et heop t ima lc on t r o l, yus Lyapunovana l i s,i nva r i an ts e tando t he r ma t ur e ys t heo r i e s,i tha sp r omo t ednew l e apsf o rp r ed i c t i ve con t r o lt heo r t udyandha sob t a i nedp l en t eousr e Ｇ ys [ ] s e a r chr e su l t s１－３ ． I no r de rt or educ et heon Ｇ l i neMPCc ompu t a t i on , [ ] bur den t het e chn i fRe f ２ wa sapp l i edo f f Ｇ queo l i ne,sucht ha tas equenc eo fne s t ede l l i o i dsand ps co r r e spond i ngs t a t ef e edba ck ga i nsa r ec ons t ruc t ed． Ther e a l Ｇ t imef e edba ck ga i ni sp r ope r l s enon Ｇ ycho l i nef r om Re f [ ４]t oRe f [ ７]．Howeve r,c ompa r ed t ot het r ad i t i ona lp r ed i c t i vec on t r o lwi de l edi n y us , i ndus t r i a lpr a c t i c e S ＧMPCha sf ewc a s e so fsuc c e s s Ｇ f u lapp l i c a t i ons．The r ea r es eve r a lva l uab l er e su l t s, sucha smu l t i Ｇmode lRMPC ( r obus tmode lp r ed i c t i ve ) con t r o l t ot hec on t r o lo fnuc l e a rs t e am gene r a t o r ands t ab l et oac on t i nuouss t i r r edt ankr e a c t o rde Ｇ [ ８－１１] v i c e si nl abo r a t o r c e s s f u l l ．Thes t a t uso f ysuc y l a cko fpr a c t i c a lapp l i c a t i oni saga i ns tt heo r i i na li n Ｇ g
t ens i ono fS ＧMPC． Thes t a r t i ngpo i n to fs t udyi nt h i spape ri st ak Ｇ i nga c t ua lcon t r o ldemando ft i c a lcomp l exi ndus Ｇ yp t r i a lp r oc e s sa s ba ckg r ound,and r e s e a r ch above r ob l ems．Fur t he rmo r e,a f t e rana l z i ngandimpr o Ｇ p y v i ng S ＧMPC me t hod,an e f f e c t i ves t r a t egy i s de Ｇ , s i i chapp l i e scomp l exS ＧMPC me t hodi na c Ｇ gned wh t ua li ndus t r i a lp r oc e s s．
１ L imeRo t a r l nPr o c e s sDe s c r i t i on yKi p
Ac t i vel imero t a ryk i l nsys t em hasmanyt e chＧ no l og i c a lproc e s s e s．Theproc e s si sma i n l d i v i d ed y i nt o ma t e r i a lsys t em and a i rf l ow sys t em．Raw ma t e r i a lbe come s produc ta f t e r pr ehea t i ng,h i gh t empe r a tur ec a l c i na t i onandcoo l i ng．Pr ima rya i r, s e conda rya i randcokeovengascons t i tut ethea i r f l ow sys t em．Se conda ry a i rc ancoo lthef i n i shed t．Exhaus tgas e sent e r pr ehea t e rand pr e Ｇ produc hea tr aw ma t e r i a l． The who l e t e chno l og i c a l e s si sshowni nF i １． I nt h i spape r,t hepr o Ｇ proc g /d, duc t i onc apa c i t fl imek i l nunde rs t udyi s６００t yo i nt hes i z eo f４ ０ m×６０ m．
Founda t i onI t em: I t em Spons o r edbyNa t i ona lNa t ur a lSc i enc eFounda t i ono fCh i na ( ６１０３４００５) B i o r aphy: ZHANG L i( １９７９—),Ma l e,Do c t o r,Le c t ur e sh i Ｇma i l:neuzhang l i＠１６３ c om; Re c e i v e dDa t e:Ap r i l２３,２０１２ p; E g
I s sue８ App r oa cho fSyn t he s i z i ngMode lPr ed i c t i veCon t r o landI t sApp l i c a t i onsf o rRo t a r l nCa l c i na t i onPr o c e s s １５ yKi
augmen t ed obs e r va t i on ma t r i x Γf ands t a t espa c e ∧
ve c t o re s t ima t i on Xf c anbeob t a i nedt hr oughSVD de compos i t i on．
３ App r oa cho fS ＧMPC
F i １ Te chno l o i c a lp r o c e s scha r to fr o t a r i l n g g yk
２ Ca l c i na t i onPr o c e s sMod e l l i ng POＧMoe sp me t hodi sus edi nc a l c i na t i onp r oc e s s mode l l i ng．Thes t a ckve c t o rs t a t espa c eequa t i oni s scu r r en tt ime i n t r oduc edi nEqn ( １)．Suppo s ek a andf a sf u t u r et ime,t hen ( ) ( yf ＝Γfx k ＋HfUf ＋GfWf ＋Vf １) éê ùú ( ) y k ê ú ê ( ) úú ê y k＋１ ú ,Γf ＝ De f i nef u t u r e ou t t yf ＝ ê pu ê ⋮ ú ê ú êê ( ) k＋f－１ úúû ë y éê éê D ０ ０ ùúú C ùúú ê ê ê ú ê CB D ０ úú ê CA ú ê ê ú , Hf ＝ ê ú, ⋮ ⋮ ⋮ ⋱ ⋮ ú ê ú ê ê ú ê ú êê êê f－１ úú f－２ C C B CAf－３B D úúû ë A û ë A o t he rs t a ckve c t o rGf ,Wf ,Vf andf u t u r ei npu tUf havet hes imi l a rpe r f o rmanc e．Al s o,pa s tt imes t a ck ve c t o rc anbede s c r i beda s: ( yp ＝Γpx( k－p)＋HpUp ＋GpWp ＋Vp ２) [ ) , ( ) ( whe r e pa s tou t typ ＝ y k－p y k－p ＋１ pu T k－１)] ando t he rve c t o r sandma t r i c e shaves imＧ y( i l a rde f i n i t i ons．Eqn ( １)andEqn ( ２)a r ewr i t t eni n : Hanke lma t r i xf o rm ( Yf ＝ΓfXf ＋HfUf ＋GfWf ＋Vf ３) ( Yp ＝ΓpXp ＋HpUp ＋GpWp ＋Vp ４) I npu tandou t tHanke lma t r i xexp r e s s eda s: pu éê y( k) y( k＋１) y( N －f＋１)ùúú ê ê ú, Yf ＝ ê ⋮ ⋮ ⋱ ⋮ ú ê ú êë ( k＋f－１) y( k＋f) y( N ) úû y whe r e Hanke l ma t r i xYp ,Uf ,Wf ,Vf ,Up ,Wp , andVp haves imi l a rde f i n i t i on．Tos o l ves t a t espa c e , mode l QR de c ompo s i t i oni s madet o Hanke l ma Ｇ t r i x． é ù é ù éê T ùú ê fú ê １１ ０ ０ ０ ú Q U L １ ú ê ú ê ú ê ê ú ê ú ê Tú Y L L２２ ０ ０ ú Q ê ２ú ê p ú ê ２１ ( ú ú ê êê úú ＝ êê ５) ê Tú U L L３２ L３３ ０ úú Q ê ３ú ê pú ê ３１ ú ê ú ê ú ê êë ú êë Y L L４２ L４３ L４４ úû êëQ４T úû ４１ f û é Tù ∧ ê ２ú Q [ ４２ L４３ ] êê T úú ＝ΓfXf ,t WhenN →∞ ,t hen L he êë úû Q３
Thef o l l owi ngs t ab i l i z ab l et ime Ｇ de l ayl i ne a rs Ｇ ys t em wi t hapo l t op i cunc e r t a i n t s c r i t i oni scon Ｇ y yde p s i de r ed: p
x( k＋１)＝A( k) x( k)＋ ΣAj ( k) x( k－ j＝１
( τj )＋B( k) u( k－τ) ６) nu nx , , r es t a t er e s e c Ｇ whe r e u∈R a r ei npu t andx∈R a p t i v e l τandτi , i∈{ １,２,,p}a r ei n t e e ri npu tand y; g |A１ ( ||Ap ( |B ( s t a t ed e l a s;[ A( k) k) k) k)]∈Ω y whe r eΩ i st heconvexs e t．As sumet ha tAj ( k),j∈ { , ,, } １ ２ r ea s t o t i c a l l t ab l eandf o rs imＧ ymp ys p a l i c i t p y,０≤τ＜τ１ ＜ ＜τp ． Thepur ei st ode s i r ed i c t i vecon t r o l l e r pos gnap , t ha tbr i ngst hes t emt ot heequ i l i br i um po i n t a t ys ch i ev i ngt hef o l l owi ngcos ti ndex: e a cht imek a mi n J∞ ( k)＝ max u( k) [ A( k＋i) k＋i) k＋i) k＋i)]∈Ω ．∀i≥０ |A１( ||Ap ( |B( ∞ ２ W i＝０
Σ [‖x( k＋τ＋i| k)‖ ＋‖u( k＋i| k)‖２R ]
k＋i| k)≤u,－ψ≤ψx( k＋τ＋ s t －u≤u( －
－
( i＋１| k)≤ψ,∀i≥０ ７) /upp whe r e,uandui si npu tc on s t r a i n t sl owe r e rbound; －
/uppe －ψandψi ss t a t econs t r a i n t sl owe r rbound;W －
andR i spe r f o rmanc ei ndexwe i tma t r i c e s． gh
３ １ Of f Ｇ l i nepr e d i c t i v ec on t r o l Toso l veEqn ( ７),t hep r ed i c t i ono fx( k＋i| k) i sne eded． I ti spos s i b l et ode f i nes e t si n wh i cht he f u t ur es t a t e swi l ll i e．De f i net hes e tχ( k＋i|k)＝Co { xli－１ l１l０ ( k＋i|k),１≤lj ≤L ,０≤j≤i－１}．As Ｇ sumi ngt ha tx( k＋i| k)∈χ( k＋i|k),i fxli－１ l１l０ ( k ＋i| k)s a t i s f i e s li－１l１l０ ( x k＋i＋１| k)＝Alixli－１ l１l０ ( k＋i| k)＋ p
li li－τj －１ ΣAj x j＝１ li li－１l１l０
l１l０
( k＋i－τj| k)＋
( ( B u k＋i－τ| k) ８) , ( ) whe r e χ k＋i|k i st het i t e s ts e tt ha tcon t a i ns gh a l lpos s i b l ef u t ur es t a t e sx( k＋i＋１|k)．Nows e l e c t Lyapunovf unc t i onr e f e r r i ngt ot he me t hodo f Ko Ｇ ha r ei n１９９６．Thes t ab i l i t t r a i n t sc anbego t: ycons T Tù é êΛ Ml Ξ ú ê ú ( êêMl Λ ０ úú ≥０, l∈ { １, ２,, L} ９) ê ú ê Ξ ０ γI úû ë , { whe r e Λ ＝d i ag Q ,Q０ ,Q１ ,,Qp },Ml ＝
J ou r na lo fI r onandS t e e lRe s e a r ch,I n t e r na t i ona l Vo l ２０
１６
l l l l éê l A Q B Y A１Q１ Ap－１Qp－１ ApQp ùúú ê ê ú ０ ０ ０ ０ Q ê ú ê ú ê ú, ０ ０ ０ ０ Ξ＝ ０ Q ê ú ê ú ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ê ú ê ú ê ú Qp－１ ０ ０ ０ ０ ë û １/２ éê ù ０ ０ú W Q ０ ０ ê ú ．I tf o l l owst ha t / êê ０ R１ ２Y ０ ０ ０úúû ë J∞ ( k)≤V [ w( k＋τ|k)]．De f i neV [ w( k＋i|k)]≤ , , γ whe r eγi sauppe rbounds c a l a r andEqn ( ９) i sequ i va l en tt o:
１ ∗ ùúú ≥０,∀l０ , l１ ,, wlτ－１ l１l０ ( k＋τ| k) Λ úúû L} ( lτ－１ ∈ { １, ２,, １０) Suppo s et he r ea r eaγ,s Z, mm e t r i c m a t r i c e s y , { , ,, , , , } Γ Q Q０ Qj j∈ １ ２ t r i xY ＝F p andma ( ha tEqn ( ９),Eqn ( １０)andt hef o l l owＧ k) Q０sucht
éê ê êê ë
i ngLMIa r es a t i s f i ed． é ù êZ ú Y ú ２ , j ≤uj êêê T １,, nu } ( １１) i ． n f, j＝ { ú ≥０ Zj Y ë Q０ úû êé Γ ∗ ∗ ∗ úù ê ú T ê ú AlQ) Q ê( ú ψ ê ú l T ú ≥０ ) ( B Y １２) êê( Q０ ψ ú ê ú ê ⋱ ú ê ú T ê( l ] Qp úû ApQp ) ë ψ whe r e,uj,inf＝ mi uj ,uj };ψs,inf ＝ mi n{ n{ s, ψ ψs}; － －
Zjji st hejt hd i agona le l emen t so fZ．Thent hecon Ｇ s t r a i n ti ss a t i s f i ed．Ateacht imeapp l i ngcont ro l y ( ) ( ) ( ) , u k ＝F k x k t hef o l l owi ngr e su l tc anbeob Ｇ t a i ned:
mi n
γ
γ, Q, Q０, Q１, , Qp , Y, Z, Γ
s t Eqn ( ９), Eqn ( １０), Eqn ( １１), Eqn ( １２) ( １３) Re f e r r i ngt ot heo f f Ｇ l i neRMPCi n Eqn ( １１),
t heo f f Ｇ l i neme t hodc anbeob t a i nede a s i l st hea l Ｇ ya r i t hm１． go １)Of f Ｇ l i n e．S e l e c ta ugme n t e ds t a t e swi ,i∈ { １, lτ－１l１l０ ,N },t ( h e ns ub s t i t u t ee a c hs t a t ew k＋τ|k) i nEqn ( １０),c ompu t et hec o r r e spond i ng { γi ,Qi , nx ( τp ＋１) T , } , { Λi Yi e l l i o i dsεi ＝ w ∈R |w Λi－１w ≤ ps －１ １}andf e edba ckga i nsFi ＝YiQi ,whe r ewi shou l d , { ,, } , s a t i s f ε ⊂ ε ∀ ∈ ２ N a n d y j j－１ j l T －１ l Λi－１ － ( Ti l∈ { １,, L} ( １４) ＋１ )Λi Ti＋１ ＞０,
２)On Ｇ l i ne．Ate a cht ime,kt ake st hef o l l owi ng s t a t ef e edba ckl aw: ïìF[ αi( k)] x( k), w( k)∈εi , ï ï ( u( k)＝F( k) x( k)＝ íï w ( k)∉εi＋１ , i≠N １５) ï ïF x( ) , ( ) k w k ∈ ε î N N whe r e,F [ αi( k)]＝αi( k) Fi ＋ [ １－αi ( k)] Fi＋１ ,and T －１ －１ w( k) { αi( k) Λi ＋ [ １－αi( k)] Λi＋１} w( k)＝１．
３ ２ Ou t t Ｇ t r a ck i ngde s i pu gn ３ ２ １ MPCf o rt rac k i ng Suppos et ha ts t emou t ty( k)i sr equ i r edt o ys pu , ( t r a ckt heg i v e nt a r e t ． A ts t e a d s t a t e z ＝ x g yr y s s, ) , ( ) us i saso l u t i ono fEqn １６ ． ( Ezs ＝Fyr １６)
éê ù ０ A０ －Inx B０ ùúú nx , ny ú ê ú, , F ＝ nd [ A０ , êê úú a C０ D０ úúû I ë ny û B０]i sthenomi na lmode l．Then,theso l ut i ono f th i sequa t i onc anbepa r ame t e r i z eda szs ＝ Mθθ and r eθ∈Rnθ i sapa r ame t e rve c t o r,and yr ＝Nθθ,whe －１ T ìï[ i fr＜nx ＋nu ï VΣ U FG V ⊥ ] Mθ ＝ íïï －１ T and V i fr＝nx ＋nu î Σ U FG ïì[ i fr＜nx ＋nu ny , nx ＋nu －r ] ï G ０ ．He r e,E ＝ Nθ ＝ íïï i fr＝nx ＋nu îG ( ) UΣVT ,r＝r ank( E ),U ∈R nx ＋ny ×r ,Σ∈Rr×r ,V ∈ ìï I i fr＝nx ＋ny ï ny ( ) ．Ma t r i x R nx ＋ny ×r and G ＝ íïï T ) ( fr＜nx ＋ny î F U ⊥ ⊥i Gi saf u l lco l umnr ankandrg ≤ny ． ３ ２ ２ De s ignofo Ｇ l i ner o bu s ts t a t eo b s e rve r ff A s imp l eLuenbe r robs e r ve rt oe s t ima t et he ge s t a t eo fEqn ( ６)i sus ed:
whe r e,E ＝
éê ê êê ë
∧
∧
x( k＋１)＝A０x( k)＋B０u( k)＋ ∧
Lp [ k)－y( k)], y( ∧
∧
( １７)
k)＝Cx( k)＋Du( k) y( ∧ nx k)∈R i st hecur r en tobs e r ve rs t a t e,and whe r ex( Lp ∈Rnx ×ny i st heobs e r ve rga i nt obede t e rmi ned．
３ ３ She du l e d Ｇ t r a ck i ng MPCde s i Al r i t hm２) gn ( go Eq n ( １)g i v e st h ee i l i b r i ums u r f a c ea ndc ompu t e s qu s ( ) t h eexpe c t edequ i l i br i um po i n tf r omyr byEqn １６ ． １)Of f Ｇ l i ne．S t a r tf r om j＝０,spe c i f st he yj a () () x j ,u j ),so l vea l Ｇ co r r e spond i ngcon t r o l l e ro f( go ( j), ( j), ( j), ( j)} , { , ,, r i t hm１ andge t γi Qi Λi Yi i＝１ () () () N ．Andt hes t a t ef e edba ckga i nFij ＝Yij ( Qij )－１i s t r o l l e ri nl ook Ｇupt ab l e．Al ongt hed i Ｇ s t o r eda sjcon ( ( s) s) ( , ) , r e c t i ono fx u choos et henex tequ i l i br i um ( ) ( ) i n t( x j＋１ ,u j＋１ )．Le tj ＝j ＋１,r epe a tt h i s po () r oc e s sun t i l ∪jM＝０ ∪iN＝０εij ,and M ＝ maxj cove r p ( ( s) s) ( , ) t heexpe c t edpo i n tx u ． ２)Gi v i ngani n i t i a lx( ０)．Le tt hes t a t ebex( k) ( ) r f o rm al i nes e a r chove r{ γij , a tt imek．Then pe ( j), ( j), ( j)} Qi Λi Yi i nt hel ook Ｇupt ab l et of i ndt hel a r Ｇ s tcon t r o l l e rj andt hel a r s tsubs c r i ti such ge ge p () () t ha tx( k)∈εij ． Imp l emen tu( k)＝Fij ( k)[ x( k)－ ( ( j)] j) x ＋u ons t em． ys
４ Ro t a r l nCa l c i n i z i ngZon eTemp e r a t u r e y Ki MPCCon t r o ls s t em y I nF i ２,t her o t a r i l ni st ake na st her e s e a r c h g yk
I s sue８ App r oa cho fSyn t he s i z i ngMode lPr ed i c t i veCon t r o landI t sApp l i c a t i onsf o rRo t a r l nCa l c i na t i onPr o c e s s １７ yKi
ob e c t,andc a l c i n i ngbe l tt empe r a t u r ep r ed i c t i vecon Ｇ j t r o ls t emi sde s i on t r o ls t emc ons i s t s ys gned．Thec ys o fs e tva l uemodu l e,c a l c i n i ngbe l tt empe r a t ur eso f t s ens o r,syn t he s i z i ng mode lp r ed i c t i vec on t r o lmod Ｇ
u l e,e t c．Asshowni nF i ３,on Ｇ l i necon t r o ls t em g ys , cons i s t so ft a r tc a l cu l a t i ngun i t op t imi z a t i onso l u Ｇ ge t i onun i t,s t a t eobs e r ve run i tandr e cur s i vesubspa c e adap t i veun i t．
F i ２ S t r u c t u r ed i a r amo fr o t a r i l nc a l c i n i ngb e l tt emp e r a t u r ec on t r o ls s t em g g yk y
F i ３ On Ｇ l i n ec on t r o ls s t emd i a r am g y g
５ S imu l a t i onExp e r imen t
t ot hes t em ma t r i xchanger angeandchangel aw ys i nt heope r a t i onp r oc e s s,i n t r oduc et hepe r t urba t i on , , , t e rmμ１ μ２ andμ３ ． So t hes t a t espa c emode lpa Ｇ r ame t e r sa r ea sf o l l ows:
I no r d e rt oc h a r a c t e r i z et h eun c e r t a i n t c c o r d i ng y,a éê ùú ( ) ０ ９ ９ ３ ６ k ０ ０ ０ １ ５ －０ ００６２ －０ ０００７ －０ ０００１ １ μ ê ú ê ú ０ ００６３ ０ ９９０６μ２( k) ０ ０１７０ －０ ００１７ －０ ０００１ ê ú ê ú ú A( k)＝ êê －０ ０００９ ０ ００２６ ０ ９７９３ －０ ０１４７ ０ ０２０９ ú ê ú ０ ００００ －０ ０００１ ０ ００１８ ０ ９９８５ －０ ００６９ ê ú ê ú ê －０ k)úû ００００ ０ ０００１ －０ ００１５ ０ ００６２ ０ ９８４８μ３( ë whe r e,μ１ ∈ [ ０ ８０,１ ２０],μ２ ∈ [ ０ ８８,１ ０７],μ３ ∈ Ts＝１０s,de l aypa r ame t e rd１ ＝３,d２ ＝１０．ConＧ [ , ] , ( ) , ( ) , ( ) [ , ] , [ ０ ９９ １ ０１ B k ＝B０ C k ＝C０ D k ＝D０ ． s t r a i n t sψ＝ ０ ０ ψ＝ １５００,１３００],u＝ [ ０,０],
I ti ssuppo s edt ha tt heunc e r t a i npa r ame t e r sa r es e Ｇ l e c t edr andoml na l l owab l er ange．Samp l i ngt ime yi
－
－
u＝ [ １２０００,５００００],we i t i ng ma t r i xW ＝d i ag( １, gh
１８
J ou r na lo fI r onandS t e e lRe s e a r ch,I n t e r na t i ona l Vo l ２０
１,１,１,１),R ＝d i ag( ０ １,０ １)． ) １ Cons t an tva l uet r a ck i ngexpe r imen t S imu l a t i onha sob t a i nedt heou t tcu r ve so f pu t he c on t r o l l ed va r i ab l e si nc l ud i ng c a l c i n i ng z one t empe r a t u r ey１ andk i l nt a i lt empe r a t u r ey２ sucha s F i ４ ( a)and ( b)．Compa r ed wi t ht he me t hodi n g Re f [ ８],t heme t hodo ft h i spape rha sobv i ousad Ｇ van t age ssucha ssho r tr i s et imeandsma l lf l uc t ua Ｇ
t i oni ns t e adys t a t e．I nf i xedva l uet r a ck i ngs i t ua Ｇ , t i on t hecompa r i soncur ve so fga sf l owands e cond Ｇ a r i ra r eshowni nF i ４ ( c)and ( d)．Ea chs t a t e ya g componen t compa r i son cur ve si n cons t an t va l ue t r a ck i ngs i t ua t i ona r ede s c r i bedi nF i ５．Fo ru s i ng g t he ga i ns chedu l ed pr ed i c t i vecon t r o l,t hecon t r o l va r i ab l echangei nl adde rshapeexh i b i t st het r ans a c Ｇ t i onp r oc e s sa te a chi n t e rmed i a t es t e adys t a t e．
( a)Ca l c i n i ngz onet empe r a t ur eou t t; ( b)Ki l nt a i lt empe r a t ur eou t t; ( c)Coa lga sf l ow; ( d)Se c onda r i rf l ow． pu pu ya
F i ４ Compa r i s onc u r v eo fc on s t an tv a l u et r a ck i ngs i t ua t i on g
F i ５ Compa r i s onc u r v eo fs t a t ec ompon en t si nc on s t an tv a l u et r a ck i ngs i t ua t i on g
２)Se r vot r a ck i ngexpe r imen t Se tt hei n i t i a lc a l c i n i ngz onet empe r a t ur eand k i l nt a i lt empe r a t u r ea r e１０００and９００ ℃． S t udyt he ab i l i t fy１ and y２ t r a ck i ng va r i ab l eg i ven va l ue． yo Thec on t r o l l ed va r i ab l e sy１ and y２ s e r vot r a ck i ng cur vei sshowni nF i ６．Ande a chs t a t ecomponent g
compa r i soncur ve sa r eshowni nF i ７． g Thes imu l a t i onr e su l t sshowt ha tf o rt het ime va r i ngk i l ncon t r o lob e c tt hecon t r o ls t r a t egyp r o Ｇ y j edi nt h i spape rc anno ton l t r o lt empe r a t ur e pos ycon t oa ch i evet hede s i r edt a r tva l ue,bu ta l soha st he ge r obus tandf a s tt r a ck i ngab i l i t r edwi t ht he y．Compa
I s sue８ App r oa cho fSyn t he s i z i ngMode lPr ed i c t i veCon t r o landI t sApp l i c a t i onsf o rRo t a r l nCa l c i na t i onPr o c e s s １９ yKi
( a)Ca l c i n i ngz onet empe r a t ur eou t t; ( b)Ki l nt a i lt empe r a t ur eou t t． pu pu
F i ６ S e r v ot r a ck i ngs i t ua t i onc ompa r i s onc u r v e g
F i ７ S e r v ot r a ck i ngs i t ua t i onc ompa r i s onc u r v eo fs t a t ec ompon en t s g
me t hodi nRe f [ ８],t hep r e s en t me t hodha sbe t t e r r f o rmanc e． pe
６ Conc l u s i on
Th i spape ri n t r oduc e sandpu t sf o rwa r das e r i e s o fnew a l r i t hmst oana l z eandimp r ovet hep r e Ｇ go y d i c t i vec on t r o l．Andane f f e c t i ves t r a t egyi sde s i gned t oapp l omp l ex p r ed i c t i vec on t r o lt op r a c t i c a li n Ｇ yc , dus t r i a lpr o c e s s．Howeve rt he r ei ss t i l lag r e a td i s Ｇ t anc ebe twe ent her e su l t sandp r a c t i c a li ndus t r i a lap Ｇ l i c a t i ons;t h i si s ma i n l c aus et hep r ob l emt ha t p ybe t hei ndus t r i a lf i e l di smo r ec omp l ext hans imu l a t i on env i r onmen t．Howt oex t r a c tp r ob l emsf r omi ndus Ｇ t r i a lpr a c t i c eandgu i det hed i r e c t i ono ft h e o r e t i c a lr e Ｇ s e a r c hh a sb e c omet h es t udyf o c u s． Re f e r enc e s: [ １] MayneD Q,Rawl i ngJB,RaoCV． Cons t r a i nedMode lPr ed i c Ｇ t i veCon t r o l:S t ab i l i t t ima l i t J]．Au t oma t i c a,２０００, yandOp y[ ３６( ６):７８９． [ ２] Ko t ha r eM V,Ba l akr i shnanV,Mo r a r iM．Robus tCons t r a i ned Mode lPr ed i c t i veCon t r o lUs i ngL i ne a rMa t r i xI nequa l i t i e s[ J]． Au t oma t i c a,１９９６,３２( １０):１３６１．
[ ３] Cuz z o l aFC,Ge r ome lJC,Mo r a r iM．AnImp r ovedApp r oa ch f o rCons t r a i nedRobus tMode lPr ed i c t i veCon t r o l[ J]．Au t oma t Ｇ i c a,２００２,３８( ７):１１８３． [ ４] WanZY,Ko t ha r eM V．AnEf f i c i en tOf f ＧL i neFo rmu l a t i ono f
Robus tMode lPr ed i c t i veCon t r o lUs i ngL i ne a rMa t r i xI nequa l i Ｇ t i e s[ J]．Au t oma t i c a,２００３,３９( ５):８３７． [ ５] Cychowsk iM T,MahonyO． Fe edba ckMi n ＧMaxMode lPr ed i c Ｇ t i ve Con t r o l Us i ng Robus t One ＧS t ep Se t s[ J]．I n t e r na t i ona l J our na lo fSys t emsSc i enc e,２０１０,４１( ７):８１３． [ ６] Kouva r i t ak i sB,Ro s s i t e rJA,SchuurmansJ．Ef f i c i en tRobus t Pr ed i c t i veCon t r o l[ J]． IEEE Tr ans a c t i onson Au t oma t i cCon Ｇ t r o l,２０００,４５( ８):１５４５． [ ７] Bao c angD,YugengX,Ma r c i nT,e ta l．ASyn t he s i sApp r oa ch
f o rOu t tFe edba ckRobus tCons t r a i nedMode lPr ed i c t i veCon Ｇ pu t r o l[ J]．Au t oma t i c a,２００８,４４( １):２５８． [ ８] WANZhao Ｇ e shV Ko t ha r e．AFr amewo rkf o rDe Ｇ yang,Mayur s i fSchedu l ed Ou t tFe edba ck Mode lPr ed i c t i ve Con t r o l gno pu [ J]． J our na lo fPr o c e s sCon t r o l,２００８,１８( ３/４):３９１． [ ９] KE Hu,YUANJ i ng Ｇ i．Mu l t i ＧMode lPr ed i c t i veCon t r o lMe t h Ｇ q odf o rNuc l e a rS t e am Gene r a t o rWa t e rLeve l[ J]．Ene r Ｇ gyCon ve r s i onand Managemen t,２００８,４９( ５):１１６７． [ １０] Gr ube rJK,Rami r e zD R,Al amo T,e ta l．Mi n ＧMax MPC
Ba s edonanUppe rBoundo ft heWo r s tCa s eCo s tWi t hGua r Ｇ an t e edS t ab i l i t l i c a t i ont oaP i l o tP l an t[ J]．J our na lo f y．App Pr o c e s sCon t r o l,２０１１,２１( １):１９４． [ １１] Di ngB,HuangB．Cons t r a i nedRobus tMode lPr ed i c t i veCon Ｇ t r o lf o rTime ＧDe l aySys t ems Wi t hPo l t op i cDe s c r i t i on [ J]． y p I n t e r na t i ona lJ our na lo fCon t r o l,２００７,８０( ４):５０９．