AN ITERATIVE METHOD FOR TUNING DECENTRALIZED PID CONTROLLERS. F. Vázquez, F. Morilla, S. Dormido

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1 AN ITERATIVE METHOD FOR TUNING DECENTRALIZED PID CONTROLLERS F. Vázquz, F. Morlla, S. Dormdo Dpto d Informátca y Automátca.Facultad d Cncas.UNED. C/ Snda d Ry s/n 8040 Madrd (SPAIN) Abstract: In ths papr a mthod of tunn dcntralzd PID controllrs for multvarabl systms s prsntd. It s basd on an tratv numrc alorthm, and t uss nformaton comn from th frquncy rspons of th full matrx of th systm transfr functons. Thn, th ffcts of th ntracton ar ncludd n th dsn bcaus th off-daonal lmnts n ths matrx ar tan nto account. Th am s to mt th dsn spcfcatons that wr an marns, phas marns or a combnaton of both n th dffrnt xprmnts carrd out. Thr xampls ar shown: two obtand from th ltratur and on from a ral plant. Th rsults ar compard wth anothr typ of MIMO tunn. Copyrth 999 IFAC. Kywords: PID controllrs, multvarabl control systms, an and phas marns.. INTRODUCTION Althouh numrous tchnqus of advancd control xst (robust, prdctv, adaptatv control, tc) th dcntralzd control s stll frquntly usd n ndustry as a straty of multvarabl control, as a mr control tchnqu or usd n th most ntrnal control loop and wth som of th prvous ons n a supror lvl. Th sacrfc that supposs th nvarabl dtroraton of th bnfts of a dcntralzd control structur whn s compard wth a full multvarabl control straty, s compnsatd wth crtan advantas l dsn and hardwar smplcty or asnss of us. Howvr, th us of ths typ of strats s not always possbl du to th ntracton ffcts. A svr ntracton wll produc a loss on closd-loop prformanc and stablty. In ths cass t wll b ncssary to v up th us of ths tchnqus and of thn n anothr typ of strats commntd prvously. In ordr to study ths ntracton ffcts numrous mthods xst, on of most oftn usd n ndustral applcatons bn th rlatv an matrx (RGA) (Brstol,966) wth thr numrous varants, du to thr calculaton smplcty. Th us of ths matrx s not only lmtd to th stady stat analyss to solv th parn problm amon varabls. Furthrmor, t s supplmntd wth othr calculatons to provd ndxs of stablty, or ntracton dynamc analyss. In ths ln, mthods l th drct and nvrs Nyqust s arrays (DNA and INA), (Rosnbroc,979; Macowsy,989), supplmntd wth Grshorn s bands, and othr calculatons l th condton numbrs, Ndrlnsy's ndx, dynamc RGA, tc, ar frquntly usd. Onc ths typ of dcntralzd tchnqus has bn slctd for us, thr bcaus th ntracton s not svr or bcaus only som of th systm outputs ar ndd, and onc th parn problm has bn solvd, th follown stp wll b choosn th controllr and ts tunn. Most of th paprs publshd n ths fld us PID controllrs (wth thr dffrnt possblts) for a multtud of wll-nown rasons: ndustral mplantaton, robustnss, mploymnt asnss... Som mthods found n th ltratur could b classfd undr th follown hadns that show th ntrst that th topc has rasd n th last yars: Tunn mthods basd on hurstc formulas: Ths formulas ndcat th drcton n whch th paramtrs hav to b dtund to compnsat th

2 ntracton ffcts whn all th loops ar closd. Th ntal paramtrs hav bn prvously calculatd usn som SISO tchnqu, that s, man a dsn wth only th lmnts of th man daonal (th ) of th systm transfr functon matrx. (Shnsy,995; McAvoy, 983) wors ar n ths ln. Dsns basd on th rlay mthod: Ths tchnqus ar abl to obtan th dffrnt paramtrs of th controllrs from th frquncy and ampltud of a mantand oscllaton, achvd wth th rlay whn closn th loops prorssvly. Numrous traton alorthms xst, l thos of (Wan t al, 996 and 997; Mnan and Kovo, 996; Halvy t al, 996; Shu and Hwan, 998) to mnton som. Tru multvarabl tunn mthods: In ths cas, n ordr to calculat th controllr paramtrs, ths mthods do not us only a part of th nformaton of th transfr functon matrx but rathr n th dsn thy alrady ta nto account th ntracton of th off-daonal lmnts. On of th most outstandn s th wor of Ho t al (996), that provds on-ln tunn formulas to obtan th PID controllrs for two by two systms. Ths s a mthod that uss Grshorn s band to provd a corrcton to th spcfcatons of phas and an marns and thn thy ntr n a SISO formulaton. Th prsnt wor could b framd wthn ths last roup. In t, an traton alorthm s proposd to achv th PID paramtrs for any multvarabl systm, payn spcal attnton to thos of two nputs and two outputs. Frst, th tunn mthod s xposd prsntn th notaton and th usd alorthm. Latr th mthod s appld to thr xampls: two tan from ltratur and on dvlopd by th authors. Chu (998) (n ths papr a study of th ntracton and stablty charactrzaton of ths typ of systms can b found). F. : Intracton of th hddn loop on th loop. u u u u a y ~ y F. : Effct of th ntracton and th fnal rsult Wth ths schm n mnd, th proposd tunn approach nstad of calculatn a controllr for (that would b whn thr s not ntracton) wll procd to tun for ~ whch alrady ncorporats th ffct of ths ntracton. Nvrthlss, th modfcaton of wll hav ts ffct on th othr loop, a nw tunn of bcomn ncssary. Ths opraton could b mad succssvly untl th producn chans n and ar small and ach loop mts th dsn spcfcatons. Th calculaton could b carrd out usn som SISO tunn mthod adaptd, as t wll b sn latr on. y y. ITERATIVE METHOD FOR TUNING DECENTRALIZED CONTROLLERS Lt suppos a MIMO systm wth two nputs and two outputs. Th us of a dcntralzd control straty has bn dcdd and th parn problm of varabls has bn solvd. It s amd to control ths systm by mans of two controllrs and. Th controllr closs a loop btwn th controlld varabl y and th manpulatd varabl u. If th controllr has alrady bn tund by som mthod and t s souht to ma th sam thn wth th, as can b sn n Fur, btwn th controlld varabl y and th manpulatd varabl u thr s somthn ls than th transfr functon, bcaus an addtv acton appars. Ths s du to th xstnc of on of th hddn loops dscrbd n (Shnsy,995). Th ffct of ths addtv acton could b rprsntd as th bloc a shown n Fur. In th sam fur, th combnaton of and a s dnomnatd ~, follown th notaton of Zhu and 3. TUNING APPROACH FOR PID CONTROLLERS Lt suppos that and ar two PID controllrs dscrbd by = Kp + + Td s T s = Kp + + Td s () Ts whr th suprscrpt ndcats th numbr of th traton, and th subscrpt th loop to whch th calculaton rfrs. Th alorthm starts wth o and o obtand xclusvly for and by mans of a SISO mthod or an arbtrary controllr that could b Kp=, T=9999, Td=0. Th ffct of ths lcton has lttl rprcusson n th numbr of tratons and n th convrnc. Th follown tratv alorthm

3 o o s proposd: from, ~ o s obtand, and from, o ~ s obtand, follown th xprssons: ~ = + a ~ = + a () a = + a + = (3) If th coupls ~ and ~ mt th dsn spcfcatons th traton concluds. Othrws th traton numbr ncrass. F. 3: Flow daram of th alorthm Th controllr s thn calculatd for th modl ~ ~ and th controllr for th modl. Wth th nw controllrs and, ~ and ~ ar rcalculatd. Wth ths data a nw traton bns. Ths procss s shown n Fur OBTAINING THE FROM THE ~ In ordr to obtan th from th ~ a lot of SISO mthods can b mployd. Howvr th calculaton of th ~ from START 0 = p + + Td s 0 T s 0 = + + Td s p 0 T s =0 o 0 o 0 ~ Do a s not mmdat, du to crtan ~ ~ and ~ mt th dsn spcfcatons? END YES ~ ~ (S scton 4) = + NO problms: th prsnc of tm dlays n th functons can ma th analytc calculaton complcatd. Not lss mportant, th varty of typs of modls that can ta plac whn man th quotnt vn n th xprsson of a mas t mpossbl to us a partcular formula amon th xstn ons for modls of frst ordr plus dlay, two tm constant plus dlay,..., (Ho t al, 996). To solv ths problms, th proposd mthod uss an adaptaton of a nrc tunn mthod allown a numrc calculaton nstad of an analytc on, that s to say, not xprssn th as a transfr functon but as th array of frquncy rspons of th transfr functon. Ths opraton mod s much asr and qucr and th ~ wll b othr rspons arrays, wthout an analytc xprsson n th form of a concrt transfr functon. An adaptaton of th analytc mthod of Morlla and Dormdo (998) has bn chosn. Ths s an xtnson of th Aström and Hälund (984) mthod consstn n a nralzaton of th Zlr- Nchols formulas to mov a pont of Nyqust s daram of th opn loop transfr functon from a poston A (n controllr's absnc) untl anothr poston B (n controllr's prsnc), as Fur 4 shows. If th controllr s an dal PID controllr vn by th follown xprsson (s) = Kp + + Td s (4) T s th dtrmnaton of ts paramtrs can b summarzd n th quatons r cos ( φ φ ) Kp ra T = ( t ( φb φa ) + 4α + t ( φb φa ) ) (6) α ωc Td = α T (7) b b a = (5) whr ω c s th frquncy corrspondn to th pont A lctd n th Nyqust s daram, r a and φ a ar th an and th anl of ths sam pont, r b and φ b th dstnaton pont B an and anl, and α th rato btwn th drvatv and ntral tm constants, that should b spcfd. φ b φ a B A ~ Ima Ral F. 4: Procss and procss plus controllr Nyqust s daram and crcl unt. Th controllr s PID and t has bn tund to mov th pont A to th pont B. ~

4 As Morlla and Dormdo (998) show, ts dsn admt thr dsn possblts: phas marn (PM), an marn (GM) and a combnaton of both. Du to spac lmtatons, only th dsn for phas marn wll b dscrbd. Frst of all, th frquncy ran ω c that admts soluton for a crtan controllr s calculatd, that s to say, th st of ponts A, pn n mnd that th procss ncludd n th alorthm s a frquncy rspons array, ~. From ths array, th pont A an and anl, r a and φ a, ar obtand. From th PM spcfcatons, φ b s calculatd. Bcaus pont B s locatd on th unt crcl, thn r b =. Wth ths data, Kp and T ar calculatd from xprssons (5) to (6) for ach pont A wth possbl soluton. In ordr to choos on amon all of th solutons, a slcton approach s ndd. For xampl, f t s souht to mnmz th ntral of th absolut valu of th rror (IAE), th maxmum rato Kp/T wll b lood for. Paramtr α can also b chosn to mnmz som othr optmzaton approach. It s also ncssary to notc that t s possbl to us any othr SISO tunn mthod whnvr t s adaptd to solv th problms commntd bfor. 5. EXAMPLES Exampl : As a frst xampl, a watr-mthanol dstllaton column proposd by Ho t al (996) s analyzd. Th systm s dscrbd by th follown transfr functon matrx.8 s 8.9 3s 6.7s + s + (8) 6.6 7s 9.4 3s 0.9s + 4.4s + If a PM=0º and a GM= for both loops on Ho's tunn formulas ar spcfd, th follown paramtrs ar obtand: Kp =0.57, T =0.7, Td =0, Kp =-0., T =.88 Td =0; whl wth spcfcatons of PM=30º and GM=3, for both loops, thy ar Kp =0.38, T =.64, Td =0, Kp =-0.07, T =4.8 Td =0. Howvr, f an analyss of ths tunns s mad th obtand rsults show that th valus ar vry far from th ntal spcfcatons: Tunn Loop Loop th PM=5.6º GM=3 PM=94º GM=. th PM=65º GM=5.3 PM=03º GM=3.8 Ths s bcaus any dsn that uss Grshorn's bands wll provd vry consrvatv tunns whn th ntracton s apprcabl. Th man rason s that n most of th cass, ths bands ar xcssvly wd and thy produc a corrcton n th ntal phas and an marns. Grshorn's bands for ths systm wth th frst tunn can b sn n th raph of Fur 5, suprmposd wth drct Nyqust array (DNA). In ths raph ~ and ~ (on contnuous ln) ar also suprmposd, both bn qut far from th band bordrs, whch shows that any dsn that uss ths bands n ts calculaton procss wll lad nvtably to xtra-consrvatv tunns. A mor xhaustv dscrpton of Grshorn's bands and drct Nyqust s arrays can b found n Rosnbroc (979) or Macowsy (989). Usn th alorthm proposd n ths wor wth spcfcatons of PM =45 and PM =45 and bnnn wth Kp =, T =9999, Td =0 y Kp =-, T =9999, Td =0 th rsults of Tabl ar obtand for ach traton. It s obsrvd that fv tratons hav bn nouh to mt th spcfcatons n th two control loops. Tabl: Rsults obtand n ach traton for xampl wth spcfcatons of PM =PM =45º It. PM PM GM GM Kp T Kp T In Tabl othr tunns ar showd, carrd out for th PI controllrs n xampl wth dffrnt sts of spcfcatons: only PM, only GM or combnatons of both. Tabl : Tunn wth dffrnt spcfcatons for xampl Spcfcatons Prformanc Control paramtrs PM PM GM GM PM PM GM GM Kp, T, Td Kp, T, Td ,.88, ,.78, ,.88, 0-0.0, 4.4, , 5, , 7.37, ,.64, , 0.6, ,.66, , 6.5, , 6.58, , 6.5, ,.49, , 3.04, 0

5 ar analyzd, th obtand rsults (PM =67º, GM =5.35 for th frst loop and PM =6.5º, GM =4.5 for th scond) ar both far from th spcfcatons, althouh not as much as n th prvous xampl bcaus Grshorn's bands ar not as wd as n th prvous xampl. F.5: DNA wth Grshorn's bands suprmposd for xampl wth th frst tunn of Ho s wor. ~ and ~ on contnuous ln and and n dscontnuous Exampl : It s a four-coupld tans systm, also proposd by Ho t al (996) whos transfr functons matrx s vn by s.8 40s 383s + 383s + (9). 80s.5 40s 8s + 8s + By mans of Ho's tunn formulas, f a PM =40º and a GM =4 for th frst loop and a PM =30º and a GM =3 for th scond ar spcfd, th controllrs hav th follown paramtrs: Kp =0.6, T =7, Td =0, Kp =0.94, T =93 Td =0. If ths tunns Usn th tratv alorthm, dsns shown n Tabl 3 can b obtand, whr a rat proxmty btwn th spcfcatons and th rsults can b obsrvd, thr n tunns for PM, by GM or combnd of PM and GM, and usn PI or PID controllrs. Exampl 3: Lastly, n Tabl 4 som tunns for th control of a hat xchanr (Morlla and Vázquz, 997) ar shown. Th procss transfr functon matrx s vn by th follown xprsson, and ts schm s shown n Fur 6. T T 0.8 = 638s s + 00 s 00 s 6000s s + Q s N 50s + (0) Du to th wdth of Grshorn s bands (as t can b sn n Fur 7), t was mpossbl to tun th controllrs usn Ho's tunn formulas. Howvr, Tabl 4 shows that th tratv alorthm allows multpl possblts, most of thm tstd latr n ral xprncs. Hr th dffrnc btwn th dffrnt tunn could b sn. Tabl 3: Tunn for dffrnt spcfcatons of xampl. Spcfcatons Prformanc Control paramtrs PM PM GM GM PM PM GM GM Kp, T, Td Kp, T, Td , 0, 0 3.5, 40, , 09, , 73, , 08, 0.39, 9, , 48, 0.4, 9, , 55, , 54, 9.4 Tabl 4: Tunns for dffrnt spcfcatons of xampl 3. Spcfcatons Prformanc Control paramtrs PM PM GM GM PM PM GM GM Kp, T, Td Kp, T, Td , 436, 0-6.3, 46, , 46, 0 -.9, 80, , 67, 0-0., 43, , 9, , 97, , 75, 0-3.7,, , 554, , 3, , 63, 0-9.9, 3, , 30, -3., 3, , 58, , 7, , 585, 0-4.5, 50, 0

6 wth ths, vry consrvatv dsns appar n th bst of th cass, snc n othr thy don't obtan a soluton. In contrast, th mthod has bn succssfully appld to tunn th PID controllrs of svral ral systms, l a hat xchanr. ACKNOWLEDGEMENTS F. 7: F. 6: Procss schm of xampl 3. DNA wth Grshorn's bands suprmposd for xampl 3 In Fur 8 a raph of th ral tmporal rspons of th two tmpraturs s shown, whr th st ponts of T (output procss tmpratur) wr chand. Th control paramtrs corrspond to a consrvatv dsn, wth th charactrstcs of th last row of Tabl T T F. 8: Tm rspons of th systm of xampl 3 undr control CONCLUSIONS In ths papr, a nw multloop controllr tunn mthod has bn prsntd. Th mthod, basd on an traton alorthm, has bn compard wth som of th xampls usd n th ltratur of MIMO tunn, achvn dsns narr to th spcfcatons. Insd th alorthm any nrc mthod of SISO controllrs tunn could b usd but wth som varaton. Th dffculty that prsnt som dsn mthods that us Grshorn's bands has also bn shown, bcaus x 0 4 Ths wor was supportd by th CICYT (Comsón Intrmnstral d Cnca y Tcnoloía) undr Grant TAP REFERENCES Aström, K. J., T. Hälund (984). Automatc tunn of smpl rulators wth spcfcaton on phas and ampltud marns. Automatca, 0, Halvy, Y, Palmor, Z., Efrat,T. (997): Automatc tunn of dcntralzd PID controllrs for MIMO procsss, J. Proc. Cont. Vol. 7 Nº, pp 9-8 Elsvr Scnc, Ltd. Ho, W.K., T.H. L, O.P. Gan (996): Tunn of multloop PID controllrs basd on an and phas marns spcfcatons. 3 th IFAC World Conrss, pp -6. Macows, J.M. (989). In: Multvarabl fdbac dsn. Addson-Wsly McAvoy,T. (983). In: Intracton analyss: prncpls and applcatons. Instrumnt Socty of Amrca. Mnan, S., Kovo, H. (996): Rlay tunn of multvarabl PI controllrs, 3 th IFAC World Conrss, pp Morlla, F., Dormdo, S. (998): Tunn PID controllrs usn spcfcatons of frquncy rspons. Intrnal Rport. Dpto. d Informátca y Automátca.UNED Morlla, F., Vázquz, F. (997): Th pasturzr plant PCT-3. Intrnal Rport. Dpto. d Informátca y Automátca.UNED Shnsy F. G. (995). In: Procss Control Systms. N.Y.: McGraw-Hll. Shu, S.J., Hwan,S. (998): Squntal dsn mthod for multvarabl dcoupln and multloop PID controllrs, Ind. En. Chm. Rs., Vol. 37, Nº, pp 07-9 Rosnbroc, H.H. (979). In: Computr-add control systm dsn. N.Y. Acadmc Prss Wan, Q., Han, C. and Zou,B. (996): A frquncy rspons approach to auto-tunn of multvarabl PID controllrs, 3 th IFAC World Conrss, pp Wan, Q., Zou, B., L, T., and B, Q. (997): Autotunn of multvarabl PID controllrs from dcntralzd Rlay Fdbac, Automatca, Vol. 33 Nº 3 pp Zhu,Z.X., Chu, M. (998): Dynamc analyss of dcntralzd x control systms n rlaton to loop ntraton and local stablty. Ind. En. Chm. Rs., Vol. 37, Nº, pp