Robust Control Toolbox for Time Delay Systems with Time Delay in Numerator and Denominator

Transcription

1 Rct Rach i Cicit a Sytm Robt Cotol Toolbox fo Tim Dlay Sytm with Tim Dlay i mato a Domiato MRE DLP Faclty of ppli Ifomatic Toma Bata Uivity i Zli a Stami 5, 76 5 Zli CZECH REPUBLIC lapa@fai.tb.cz btact: - Th aim of thi pap i to pt Matlab toolbox fo obt cotol of tim lay ytm with tim lay i mato a omiato of th cotoll plat. Th toolbox olv th poblm of ctai tim lay ig D- itatio a algbaic appoach. Th algbaic appoach pol placmt tchiq i combiatio with voltioay algoithm Difftial Migatio a tct igla val a th obt pfomac a tability iicato. Fo tig p th fial cotoll l-ma implx mtho ca b i th cop of th toolbox. a altativ to th algbaic appoach D- itatio i ppot ptig a taa tool fo obt cotol ig. y-wo: - Tim lay ytm, obt cotol, algbaic appoach, tct igla val, ctaity Itoctio Tim lay ytm with tim lay i mato a omiato i a w i i cotol thoy. Th poblm wa itoc i [9] a [] a olv ig algbaic thoy i th ig of qai ta momophic fctio. I thi pap a toolbox i pt olvig th poblm via mltiplicativ ctaity a th poc how i [3]. Thi poc ca t a cotoll that gaat obt tability a pfomac fo th whol ag of tim lay. th aitioal tool algbaic appoach i ( [] a [3]) implmtig pol placmt tchiq a voltioay algoithm Difftial Migatio ( []) tatig th poblm of mltimoality of th cot fctio. Pol placmt i pfom via olvig th Diophati qatio i th ig of Hwitz-tabl a pop atioal fctio (R p ). Th tct igla val ot µ ( [8]) a th obt tability a pfomac of th cotoll. a altativ mtho th D- itatio ( []) ca b i th cop of th toolbox. Etopy, LMI o DF fomla a th optio i th D- itatio pat ( [5], [6] a [7]). τ [, T ], τ [, T ] () Thi family of plat ha ctai ta qaipolyomial i th omiato. Th lay vay i th itval of zo to a pfi val ptig th pp bo fo ach tim lay. Thi t of plat i tat via LFT ig th chm i Fig.. Th wight W l a W l a obtai fom th iqaliti: W li jωt i >, i, () Th ptbatio matix ha th fom: δ l l, δl δ l <, δ l <, δ l, δ l C (3) a pfomac wight i a 3 o taf fctio: b b b W 3 a a a a () 3 b a δ l W l Poblm Fomlatio Th poblm to olv i gal t with ctai tim lay: τ b P ( ) a τ o ytm Fig. LFT mol of plat b a δ l W l P om ISB:

2 Rct Rach i Cicit a Sytm Th wight W l a W l hol atify () with vy low covatim ( Fig. a 3). i th low lia factioal tafomatio o galiz plat a cotoll ( Fig. ). l - W P om Fqcy (a/) Fig. Bo plot W l (ah) a th ight i of () (oli) Fig. Clo-loop itcoctio fo µ-ythi l z w - - v -3 Fig. 3 Bo plot W l (ah) a th ight i of () (oli) 3 Poblm Soltio 3. Robt ig Stct Sigla Val Famwok Th poblm fi i pvio ctio ca b olv ig itcoctio i Fig.. H ot th galiz plat patitio to (5) wh th block tct of copo with th ipt a otpt vaiabl i Fig. 5: z w (6) v Th ig objctiv i to fi a tabilizig cotoll ch that p ~ [ F l (, )] (7) µ ω Fqcy (a/) i miimal, wh M Fl (, ) ( ) (8) Fig. 5 Clo-loop itcoctio 3. lgbaic ppoach Th plat fo which th cotoll i iv i th omial ytm: b P ( ) (9) a omial plat P ca b tafom to: b α B P ( ),, B R PS () a α Th cotoll i obtai a a oltio to th Diophati qatio M B () with BIBO tabl fback cotoll /D giv by t t T ( α ( α3 ( α () D D BT t t B ( α ( α3 ( α Th omiato of () i iviibl by o that aymptotic tackig fo th tpwi fc igal ca b achiv. ISB:

3 Rct Rach i Cicit a Sytm Th aim of ythi i to ig a cotoll which atifi coitio p µ [ FL (, )( ω, α,, α, t, t )], ω (, ) (3) ω tabilizig Th cotoll ha th fom: ( ), 3,3,, () I o to ovcom th poblm of o-itgatio tct of th D- itatio cotoll a chm with itgato that icopoat th itgatio popty ito th cotoll wa ( Fig. 6). Th cotoll ha th taf fctio: 5,5, D ( ) (5) 5 W, /, l Pom *. Sytm Dfiitio Sytm fiitio ha th btto fo iplayig th ialog fo tig paamt of th cotol plat. H th paamt of taf fctio a th maximm val of tim lay ca b t (Fig. ). Fig. Dialog fo tig paamt of th cotol plat xt two btto iplay th ialog fo tig th paamt of th wight W l a W l tatig ctai tim lay τ a τ (Fig. 3). I th ialog i a btto fo howig th Bo plot of th wight W l a W l compa to th lft ight of () ( Fig. ). Fig. 6 Clo loop itcoctio with itgato caca U Itfac Th mai wiow of th toolbox coit of th pat ( Fig. ): - Sytm Dfiitio - Cotoll Dig - Simlatio a Vificatio Fig. 3 Dialog fo tig th paamt of th wight W Fig. Th mai wiow Fig. Bo plot of th wight W compa to th lft i of () ISB:

4 Rct Rach i Cicit a Sytm I th lat pat of ytm fiitio btto howig ialog fo tig paamt of th pfomac wight W i plac. Th wight i th am fo th D- itatio a algbaic appoach. Th ialog ha a btto fo howig th Bo plot of th wight.. Cotoll Dig Th cotoll ig pat i ivi ito two ctio D- itatio a algbaic appoach. I th fit ow th a th btto fo tig paamt fo both th D- itatio a algbaic appoach. I th co ow th a btto fo pfomig th ig of cotoll. Th ig i itactiv a th comma li wiow of th Matlab ytm fo commicatio with. Th D- itatio ak th fo th tatig m-itatio. Th th fit gamma fo th boptimal cotoll i ach ig th bictio mtho. Th th i pompt fo th chag of fqcy ag a bo o tolac. Th th ct tp of th D- itatio i fiih a th µ-plot i iplay. I th xt tp th µ-plot i appoximat ig calig matic D a D -. To thi ffct th i pompt fo hi choic. Comma apf ca b fo ato-pfit, which atomatically fi th paamt fo thi tp. ft xitig thi pat ig comma paamt fo gamma ach ca b t. Th th i agai pompt fo chag of th fqcy ag a bo o tolac. Fially, th µ-plot i calclat a iplay. Th tp a pat til th tmiat th whol poc. Th th ltig cotoll i obtai a iplay i th Matlab wiow. Th algbaic appoach lach th voltioay ach pfomig th pfi mb of migatio loop fi i th paamt ialog fo th algbaic appoach. Th ach ca b itpt by pig CtlC. Th cotoll ca b obtai by pig ppoximat a gt cotoll btto. Bi th voltioay ach l-ma implx mtho ca b fo th t p of th cotoll by pig th btto T p with implx mtho..3 Simlatio a Vificatio Simlatio a vificatio pat ha two colm of btto ach fo th paticla ig mtho, i.. D- itatio a algbaic appoach. I th fit ow btto fo iplayig th µ-plot a pt. If th M-plot btto i th algbaic appoach i p th a compaio of both appoach ca b viw i tm of th µ-plot fo both th D- itatio a algbaic appoach i o fig. I th co ow th µ-plot btto fo pfomig imlatio i Matlab Simlik a plac. Th imlatio ca b pfom fo both impl fback loop a two-g-of-fom (DOF) fback loop ( Fig. 5 a 6). Fially, btto fo howig th imlatio i o plot a at th bottom of th mai wiow. If th btto fo algbaic appoach i p th th imlatio fo D- itatio i iplay i th am plot fo compaio. xampl of imlatio i o plot fo th algbaic appoach a DOF fback loop i i Fig. 7. M Fig. 5 Simpl fback loop Fig. 6 DOF fback loop D D R - Fig. 7 Simlatio i o plot fo algbaic appoach a compaio with D- itatio Simlatio P B Rfc Otpt - alg. appoach Cotol igal - alg. appoach Otpt - D- itatio Cotol iga - D- itatiol Tim (co) y y ISB:

5 Rct Rach i Cicit a Sytm 5 Dowloa Th toolbox ca b owloa fom: 8_&lagc&typ Coclio Robt Cotol Toolbox fo Tim Dlay Sytm with Tim Dlay i mato a Domiato ha b pt i thi pap. Th poblm fiitio a th ig mtho w cib. U itfac a optio of th toolbox w bifly otli. Fially, a xampl of imlatio compaig th algbaic appoach a D- itatio wa giv. ckowlgmt Thi wok wa ppot by th Eopa Rgioal Dvlopmt F th Pojct CEBI-Tch o. CZ..5/../3.89. Rfc: [] M. Dlapa, Difftial Migatio: Sitivity alyi a Compaio Sty, Pocig of 9 IEEE Cog o Evoltioay Comptatio (IEEE CEC 9), pp , 9, ISB [] M. Dlapa, R. Pokop a M., Bakošová, Robt Cotol of a Two Tak Sytm Uig lgbaic ppoach, EUROCST 9, pp , ISB [3] M. Dlapa a R. Pokop, lgbaic appoach to cotoll ig ig tct igla val, Cotol Egiig Pactic, Vol. 8, o., pil, pp , ISS [] J.C. Doyl, Stct ctaity i cotol ytm ig, Pocig of th IEEE Cofc o ciio a cotol, pp. 6-65, 985. [5] J.C. Doyl, P.P. hagoka a B.. Faci, Stat-pac oltio to taa H a H cotol poblm, IEEE Taactio o tomatic Cotol, Vol. 3, o. 8, pp , 989. [6] P. ahit a P. pkaia, lia matix iqality appoach to H cotol, Itatioal Joal of Robt a olia Cotol,, -9, 99. [7]. lov a J.C. Doyl, Stat-pac fomla fo all tabilizig cotoll that atify a H om bo a latio to ik itivity, Sytm a Cotol Ltt, Vol., pp. 67 7, 988. [8]. Packa a J. C. Doyl, Th complx tct igla val, tomatica, Vol. 9(), pp. 7-9, 993. [9] P. Zítk a J. Hlava, iochoic Ital Mol Cotol of Tim Dlay Sytm, Cotol Egiig Pactic, Vol. 9, o. 5, pp. 5-56,. [] P. Zítk a V. ča, lgbaic ig of aiochoic cotoll fo tim lay ytm, Itatioal Joal of Cotol, Vol. 76. o. 6, pp. 95-9, 3. ISB: