PART - 4. Multivariable Control for MIMO processes

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Miles McKinney
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1 PAT - 4 NU/EECS/ELEC 8500 for MMO proc Otlin - Modl 5.4 Dcoplr Din for MMO proc dal Dcoplr» Simplifid Dcoplr» nralizd Dcoplr Limitation of Dcoplr Simplr Dcoplin» Partial Dcoplin» Stad-tat Dcoplin Effct of MV Contraint ll Conditiond Proc» Dnrac» Sinlar Val Dcompoition» Dcoplin Bad on SVD

2 Mlti-loop. mlti-loop - of ral inl-loop controllr (.., PD) on pair of maniplatd/controlld ariabl mltiariabl - mak control adjtmnt dciion jointl conidrin all otpt imltanol Mlti-loop control confiration ar tpicall d a a ba control confiration and rid in th Ditribtd Control Stm (DCS).».., flow control, tmpratr control, prr control Mltiariabl control confiration tpicall rqir additional comptational capabilit, and it or a ba mlti-loop control confiration, ndin tpoint to th mlti-loop controllr. Mlti-loop. Undr th mlti-loop control trat, ach controllr ci oprat accordin to: i ( ) ci di Mltiariabl controllr mt dcid on i, not in onl ε i, bt in th ntir t, ε, ε,,...,ε n,;. Th, th controllr action ar obtaind from i ci ε i n f f f f n ( ε, ε, ε ) ( ε, ε, ε ) ( ε, ε, ε ) ( ε, ε, ε ) n n n n,p,p Mltiariabl Controllr () () () () 4

3 Principl of Dcoplin Main loop,,, n n, coplin dirabl for control Cro-coplin, i j (i j) ndirabl; loop intraction Eliminat th ffct of th ndird cro-coplin impro control prformanc. Objcti i to compnat for intraction b cro-coplin not to liminat th cro-coplin; impoibilit, rqir altrin th phical natr of th tm. 5 Simplifid Dcoplin Two compnator block and. Controllr otpt and, actal control on th proc and. Withot th compnator, and, and th proc modl Compnator, Loop informd of chan in b, i adjtd. Th am for Loop 6

4 Din Simplifid Dcoplr ( ( ) ) ( ( ) ) ( ( ) ) 7 Difficlti for Simplifid Dcoplr x, ix compnator. larr than x, dcoplin bcom tdio. NxN: (N -N) compnator. 8

5 nralizd Dcoplin MMO proc To liminat intraction, to : a diaonal matrix; (). () ( ) Choo ch that ( ) Slctd to proid dird dcopld bhaior with th implt form A commonl mplod choic ( ) Dia[ ( )] 9 lation Btwn th Two Schm Simplifid dcoplin x and x tm, th compnator tranfr fnction matrix: For th dird, tak i to find ij to mak diaonal nral dcoplin Final diaonal form pcifid a, thn can b drid. 0

6 Exampl Ditillation Colmn ) ( ) ( ) ( ) ( ) ( implifid dcoplr actal implmntation ) ( ) ( 7 ) )(4.4 )(6.7 )(0.9 (.0 ) )( (6.7 ) )( ( ) )( (.0 ) )( (6.7 ) )( (.0 ) )( (.0 ) )( (6.7 ) )( (6.7 ) )( (.0 ) )( (6.7 ) )( ( nralizd dcoplin: Exampl Ditillation Colmn Th actal implmntation:

7 Comparion of th Two Mthod Simplifid dcoplin: qialnt opn-loop dcopld tm (4.4 ) (6.7 ) 9.4(.0 )(0.9 ) (6.7 ) (4.4 ).8(.0 )(0.9 ) mch mor complicatd than pcifid in th nralizd dcoplin Difficlt to tn controllr nralizd dcoplin: tnin and prformanc bttr than for Simplifid dcoplin complicatd dcoplr Limitation in Application Prfct dcopl if modl prfct - impoibl in practic. Th implifid dcoplin imilar to fdforward controllr ralization problm, tim dla lmnt Prfct dnamic dcoplr bad on modl inr. can onl b implmntd if inr caal and tabl. x compnator, and mt b caal (no α trm) and tabl tim dla in mallr than tim dla in tim dla in mallr than tim dla in and no HP zro and mt no HP pol 4

8 mplmntation Addin dla to th inpt,,..., n, b dfin: m D d D( ) 0 d Simplifid dcoplin: rqirin th mallt dla in ach row on th diaonal, dind b in m. nralizd dcoplin: modifid proc m o that (D) - ar caal which rqirin that - (D) ha th mallt dla in ach row on th diaonal. 0 d nn 5 Exampl: Ditillation Colmn add a tim dla of mint to th inpt : ( ) Smallt dla in ach row i not on diaonal, implifid dcoplin compnator bcom: (6.7 ).48.0 Din D() to add a tim dla of mint to th inpt, i..: D( ) 0 m D (6.7 ).48.0 (4.4 )

9 Exampl: Ditillation Colmn ( ) (6.7 ).48.0 A tim prdiction trm mch mall than tim contant, drop prdiction (6.7 ).48.0 Effcti tim contant of and ar imilar Stad-tat dcoplin.48 7 Partial Dcoplin Conidr partial dcoplin if om of th loop intraction ar wak om of th loop nd not ha hih prformanc Partial dcoplin focd on a bt of control loop intraction ar important, and/or hih prformanc control i rqird. Conidr partial dcoplin for x or hihr tm main adanta: rdction of dimnionalit. 8

10 Partial Dcoplin Exampl rindin circit anali Lat niti ariabl Mot intraction: Loop and, Dcoplr: loop and, Loop withot dcoplin. th tranfr fnction matrix for th btm in th implifid dcoplin approach (7 ) (47 ) ; Stad-Stat Dcoplin Stad-tat dcoplin: tad-tat ain of tranfr fnction x tm Simplifid tad-tat dcoplin, nralizd tad-tat dcoplin Vr a to din and implmnt, firt tchniq to tr; idal dcoplr onl if dnamic intraction pritnt bi prformanc impromnt with r littl work or cot mot oftn applid in practic. 0

11 Exampl Ditillation Colmn Simplifid tad-tat dcoplin , ( ) nralizd tad-tat dcoplin Effct of npt Contraint Alwa xitin contraint on th proc inpt ariabl al cannot o bond fll opn or fll ht hatr cannot o bond fll powr or zro powr, tc. Dcoplin ok, if controllr otpt not rachd contraint En on inpt rach a contraint, (rt windp) control tm no lonr fnction dcoplin xtrml poor (or n ntabl) rpon

12 npt Contraint Exampl Simplifid tad-tat dcoplr for th WB Ditillation Colmn with c 0.0, /τ 0.07, c , /τ , f, 0 0.5, th clod-loop rpon i r poor onc th rflx al i fll opn and th tm bcom ntabl. Clod-loop rpon of Y and Y. Uncontraind maniplatd ariabl,,, pon of Y and Y with contraind, 0 0.l5. Maniplatd ariabl, whn Snitiit to Modl Error : tm tad-tat ain matrix: nralizd dcoplr - rror in th timat of th tad-tat ain matrix, thn ( ) ( ) Adj() 4

13 A and Modl Error Adj() f r mall, it rciprocal will b r lar Small modlin rror will ca r lar rror in Small chan in controllr otpt rlt in lar rror in Dcoplin difficlt: inpt/otpt ariabl ar paird on r lar A al; tm will alo b r niti to modlin rror. λ ij ij C ij C ij cofactor of ij 5 Exampl Ha Oil Fractionator Tranfr fnction modl ( ) dtrminant r clo to zro: dcoplin r difficlt Dcoplin xtrml difficlt A mall al of ain matrix dtrminant lar al of A lmnt 9.8 Λ

14 SVD and Condition Nmbr Th matrix i aid to b inlar, if it dtrminant i zro, Nar inlarit matrix: inlar al σ * / i ( λi ( A A)) i,,..., n Condition nmbr: Th ratio of th lart and mallt inlar al σ k σ max min Exampl: Ha Oil Fractionator contind inlar al σ 5.978, σ and a condition nmbr: κ clarl indicatin rio ill-conditionin. 7

[ ] 1+ lim G( s) 1+ s + s G s s G s Kacc SYSTEM PERFORMANCE. Since. Lecture 10: Steady-state Errors. Steady-state Errors. Then

[ ] 1+ lim G( s) 1+ s + s G s s G s Kacc SYSTEM PERFORMANCE. Since. Lecture 10: Steady-state Errors. Steady-state Errors. Then SYSTEM PERFORMANCE Lctur 0: Stady-tat Error Stady-tat Error Lctur 0: Stady-tat Error Dr.alyana Vluvolu Stady-tat rror can b found by applying th final valu thorm and i givn by lim ( t) lim E ( ) t 0 providd