rprns of h 8h IFAC World Congrss lano Ial Augus 8 - Spmbr obus dcnralzd conrol wh scalar oupu of mulvarabl srucurall uncran plans wh sa dla Elzava arshva Absrac h problm of a robus conrol ssm dsgn for nrconncd ssms wh srucural and paramrcal uncran was solvd for h cas whr drvavs of npu and oupu paramrs canno b masurd h ordr of h mahmacal modl ma chang ovr m Oprabl of h dsgnd conrol ssms n h cas of non-masurabl and boundd dsurbancs acng on h conrolld plan was dmonsrad Onl h masurabl varabls of h local subssms ar usd o gnra h conrol acons ha s conrol s compll dcnralzd I INODUCION E problm of conrol wh scalar npu and oupu has bcom on of h classcal problms of modrn conrol hor and pln of mhods for robus conrol dsgn hav bn dvlopd h dvlopmns n robus conrol hor as wll as a comprhnsv bblograph can b found n [ ] In monograph [] h classfcaon of dsurbancs of varous ps and wo mhods of hr compnsaon ar gvn Undr h frs approach h srucur and paramrs of h conrollng ssms ar chosn n such a wa ha h would provd nsnsv of h ssm o dsurbanc nvaran ssms h scond approach s basd on a dnamc compnsaon of nrnal and rnal dsurbancs whn h conrol of adusng a dvc supprsss h nflunc of dsurbancs on h paramrs of h ssm In addon h sud [] gvs a gnral samn of h problm and proposs a fw mhods of dsgnng h nvaran ssms ha ar basd on h algbrac srucur of mahmacal modls of plans In [-] an nrnal modl of dsurbancs s usd o solv h problm whras [6 7] us h mhods of h hor of robus and adapv ssms h approach o h snhss of sac robus conrollrs for lnar ssms ha s basd on h lnar-uadrac problm ha s n urn basd on h paramrzaon of Lur cca uaons s prsnd n [8] obus ssms wh compnsaon of dsurbancs ha us hs mhods ar sudd n [9 ] A smpl robus conrol algorhm ha rmans h sam for varous ps of plans s proposd n [] I s shown ha h algorhm compnsas for paramrc and rnal dsurbancs wh a gvn accurac A closd ssm wors hr as an mplcl gvn nomnal modl whos paramrs ar usd n conrol An ncrasng var of challngng appld problms n h adapv dcnralzd conrol hor forcs h rsarchrs o wor wh h plans wh uncran paramrs m dla and ohr ssus whch hav o b an no accoun whn dsgnng conrol ssms h faur of hs wor was suppord b h ussan Foundaon for Basc sarch proc no 9-8-7 h conrol ssms for dla ssms [-6] s h dpndnc of h sa of h conrolld procss on h prvous sas hsor Ignorng h nflunc of ha dla ma lad o h ual dgradaon of h conrol ssm and mor ovr o nabl o prform conrol funcons I s mporan o no ha almos all h suggsd mhods ar basd on an assumpon ha h srucur of a plan s nown h ordr of a ssm of dffrnal uaons s nown and paramrc and rnal dsurbancs ar unnown hr ar varous suds dvod o h problms of conrol wh an unnown ordr [7-9] Sourcs [7 8] consdr conrol problms of lnar saonar ssms wh an unnown and consan ordr of numraor and dnomnaor for hr ransfr funcons Sourc [9] consdrs a wdr class of ssms wh dsurbancs ha ar abl o nflunc boh h paramrs of h ssm as wll as s ordr hs papr consdrs h problm of robus conrol for nrconncd sa dla ssms wh unnown paramrs whch ar subc o h unconrolld rnal and paramrc dsurbancs hs dsurbancs ma chang h ordr of a ssm n unprdcabl was hs mans ha h ordr of a ssm s unnown and scalar npu and oupu sgnals can onl b masurd o solv h problm a smpl robus conrol algorhm s proposd ha compnsas for hs class of uncrans wh a gvn accurac and a fn m Onl h masurabl varabls of h local subssms ar usd for h conrol conrol s compll dcnralzd II OBLE SAEEN L us consdr an nrconncd ssm whos local subssms dnamc procsss ar dscrbd b h followng uaons G τ u f S whr d d dffrnal opraor; n n n G g n K g n K n m m rm r m K r n n S sn s n s K lnar dffrnal opraors wh unnown paramrs; f an unconrolld dsurbanc; u a scalar conrol acon; a scalar conrolld varabl n h -subssm whch can b masurd Coprgh b h Inrnaonal Fdraon of Auomac Conrol IFAC 89
rprns of h 8h IFAC World Congrss lano Ial Augus 8 - Spmbr Dcnralzd conrol for such a ssm s dfnd as h problm of fndng local conrol blocs ach of whch onl can accss currn nformaon abou a ssm [] urd ual of ranson procsss n a subssm s dfnd b uaons of h local nomnal modls m m mr r m ar lnar dffrnal opraors; m > ; r ar h scalar boundd conrol acons I s ncssar o dsgn a conrol ssm for whch h followng condon wll b sasfd lm lm < δ f r δ s h accurac of h dnamc rror ; s h m bond of whch h dnamc rror should no cd h valu δ I s forbddn o us masurabl paramrs of on subssm n ohr local subssms Assumpons m ar urwz polnomals compl varabl n Laplac ransformaon; h opraor s sabl rval soluon of uaon u s asmpocall sabl For h fd valu polnomal s urwz; h ordrs of polnomals dg n dg m dg S n n < n ar unnown and rlav dgr of a local subssm n m > ; v h uppr bound u of rlav dgr s nown as wll as h uppr bound of h dgr of h polnomal n n ; v h ordr of h polnomals m s ual o u ; v h coffcns sgns ar nown and > ; v h opraors coffcns G S ar boundd funcons; h non-zro coffcns of hgh ordrs of opraors and ar posv funcons; v h coffcns of dffrnal opraors dpnd on vcor of unnown paramrs ξ Ξ whr Ξ s a nown boundd s; h acons r ar boundd funcons; h sgnal of local nomnal modl m and s drvavs u ar boundd funcons; h rnal dsurbanc f s a boundd funcon of m wh an unnown changs rang; s prohbd o us h drvavs of sgnals u r Basd on h assumpons s possbl o conclud ha h dnamc ordr of h ssm s unnown and subc o chang as h rsul of paramrc dsurbancs For nsanc f n and n hn dg n ; f n and n hn n dg n c h rurmn o now h sgns m of h non-zro coffcns of hgh ordrs of opraors assumpon v s rlad o nowng h sgn of a hgh-frunc gan of h ssm III EOD OF SOLUION L us wr h opraors as whr s an arbrar lnar dffrnal opraor such as ha polnomal s urwz polnomal n dg hn s h dffrnc and dg n f dg < dg hn dg dg and f dg dg hn dg n s an arbrar lnar dffrnal opraor dg n u such as ha polnomal s urwz gardng srucur s possbl o sa ha f m < n u hn dg n u and f m > n u hn dg m hus s alwas possbl o guaran corrcnss of h mnond dcomposon of h opraors as n on cas opraors and hav all coffcns non-zro n ohr cas h corrspondn numbr of componns ar nonzro h dcomposon [9] ha allows o solv h problm dffrs from nown mhods of paramrzaon uaons of conrol plans L us ransform h uaon of a ssm u u G τ f S snc opraors and ar arbrar w can choos hm n ordr ha h followng condon s obd m L us wr h uaon for rror m subracng from and ang no consdraon m u u f mr G S τ 6 o oban h man rsul l's us h approach [] whch allows o compnsa dsurbanc L choos a local conrol law n h followng form u α ϑ 7 9
rprns of h 8h IFAC World Congrss lano Ial Augus 8 - Spmbr whr α > ; ϑ s an addonal conrol acon hn h followng uaon of rror can b drvd from 6 ϑ ϕ 8 ϕ m u G τ f mr S α ϑ Sgnal ϕ conans all componns acon of whch n h rror nds o b compnsad I s ncssar o rac h sgnal L s dfn h addonal loop m ϑ and wr h uaon wh h rror sgnal ζ m ζ ϕ If h drvavs u of h oupu sgnal can b masurd hn dfnng h varaon law of h addonal conrol acon n h followng form ϑ m ζ ϕ w wll g h followng uaon of h closd loop ssm usng h rror uaon 8 m L us show ha all h sgnals n h closd loop ssm ar boundd I s ncssar for h ffcnc of h algorhm whch wll b dscrbd lar Euaon shows ha h sgnal and s drvavs u ar boundd du o assumpon hn from condons of h assumpons dg n and bcaus s urwz polnomal of dgr w can conclud ha n u ϕ f m r G τ S s a boundd valu I s ncssar o show ha h chosn conrol acon s boundd For ha purpos l s subsu ϕ n wh h samn abov and rsolv drvd uaon for ϑ ϑ ϕ u α L us subsu ϑ n uaon 9 and rsolv for u ang no consdraon followng paramrzaon u ϕ From condon of assumpon and bounddnss of ϕ bounddnss of local conrol acon u s followd 9 Bcaus w canno masur h drvavs l s formula h local law of addonal conrol acon ϑ n h followng form ϑ g mζ whr g m [ m K m ] vcor composd wh u polnomal coffcns u u K ; m m ζ ζ ζ ζ col K u m u ζ ; ζ s smaon of ζ drvavs oband from flrs z ζ F z L z b ζ u u Whr ; L [ ]; b [ K ]; z K L O F O O O ; L L > s small numbr If w us and n Laplac ransformaon w ll g h followng m ϑ ζ u ang no consdraon and samn for rror sgnal ζ w hav m ϑ u Subsung ϑ n uaon 7 wh h oband samn and usng h orgnal of Laplac ransformaon w ll g conrol algorhm Obvousl ha conrol law now s chncall fasbl snc conans onl nown or masurabl varabls roposon If assumpons - ar obd hn hr ar numbrs > > such ha undr condons conrol algorhm u u α 6 m guarans ha arg condon s obd whr α > I s ncssar o no ha h dscrbd algorhm rmans nvaran f hr s sa dla n a ssm as wll as n h cas whn a ssm s n a sad sa wh unnown paramrs wh nown boundars roposon proof L s consdr vcors of h smaon rror of drvavs ζ z F b ζ u r h vcor F b h has frs componn ual o - If o prov ha h valu s small hn from condon ζ ζ < follows ha smaon ζ s 9
rprns of h 8h IFAC World Congrss lano Ial Augus 8 - Spmbr ζ nar o From w ll g h uaon of dnamc for vcors F z F h L b ζ F ζ b u ζ ang no accoun ha h addonal conrol acon s formulad as w can ransform h uaon of rror no h followng form m m 7 K ; whr m [ m m ] u col K ; ζ ζ L s ransform uaon 7 no vcor-mar form As a rsul w ll g h followng uaons s of h closd loop ssm Am b m L F h ζ 8 L u whr W v go sngularl prurbancd ssm as small nough numbr L us us Lmma [6] Lmma [6] If a ssm s dfnd b h uaon m f whr f s a connuous funcon ha s Lpshs funcon wh rspc o and n h cas whn has a boundd closd rgon of dsspaon Ω { F < C} whr F posv dfnd connuous pcws smooh funcon hn hr s > such ha undr h nal ssm has h sam dsspav rgon Ω f for som numbrs C and for followng condon s obd F sup f C f F C 9 In h cas of n 8 w hav asmpocall sabl ssm for varabls and snc A m F ar urwz mars I s h sam suaon whch w had for masurng h drvavs lm I was provd u ha f hs condon s obd all h sgnals n h ssm ar boundd I mans ha hr s a cran rgon Ω ζ { ζ δ < δ < δ F < C } whr sgnals ζ ar whn hr boundars for som nal condons from Ω L us consdr wo vcors u u θ ζ K ζ [ K ] and bloc-dagonal mars wh u dagonal blocs F dag{ F F K F } B dag{ h h K h } C dag{ L L K L } hn uaons 8 wll a h followng form Am b m L F Bθ C Evdnl ha condon 9 was obd f o a Lapunov funcon for F whr h posv dfnd smmrc mars drmnd from uaons soluon A m A F F m u I I u u ar whr hus > > > > > n accordanc wh Lmma [6] hr s such ha f < hn Ω rmans dsspav rgon of ssm 8 owvr s ncssar o no ha png h dsspav rgon dosn guaran ha h s of aracon Ω rmans h sam n a sngularl prurbd ssm L us calcula h full drvav of funcon on ssm s racors ang no accoun uaon and assgnng L us us smaons b m b Bθ m mn mn mn mn Bθ 9
whr u m B C b δ ; mn ar h mnmal and mal characrsc numbrs of h mnond mars Usng hos smaons no w ll g whr mn mn mn If o choos from condons > > h followng nual s corrc If w solv h nual w can s ha f o choos small nough w g h followng rgon of aracon Ω Fg a Srucural schm of local conrol ssm Fg b Srucural schm of robus conrol ssm Insrng h rurd valu from h arg condon no h rgh par and ang no consdraon h nuals mn mn w g h smaon of h valu δ n h arg condon mn δ ha shows ha hr ar numbrs and guaranng ha arg condon wll b obd hus for varng n and w can g h rurd valu δ n h arg condon Srucural schm of h dsgnd conrol ssm s shown n Fgur h drawbac of h proposd algorhm s a lac of analcall provd choc of paramrs and α owvr h can b asl machd durng h modlng phas For a ssm mnmall possbl coffcns of opraors G S ar usd and mall possbl valus of r f ar usd for h npu Consan componns don mar Numbrs and α ar slcd n ordr o guaran a gvn dnamc rror Numbr s usuall varng whn o Error wll no cd a gvn valu for ohr paramrs valus and valus of rnal acons from gvn class of uncran I EXALE L us consdr a dnamc ssm of sh ordr rprsnd as wo subssms sn sn cos 6 cos f u cos sn cos cos f u whr and ar h sa vcors of h subssms and ar h masurabl scalar oupus of h subssms u and u ar h scalar conrol acons whos law of varaon s gnrad accordng o E 6 and f sn f sn rprns of h 8h IFAC World Congrss lano Ial Augus 8 - Spmbr 9
rprns of h 8h IFAC World Congrss lano Ial Augus 8 - Spmbr ar h dsurbancs h paramrs of h local rfrnc m and modls ar an as h rfrnc sgnals r and r ar as follows r sn r sn W rprsn h consdrd plan usng whr h class of uncran s dfnd b h nuals ; l l l h conrollr 6 consss of wo cascadd blocs wh h followng ransfr funcons W W and amplfr wh gan of α Gvn ha δ n h arg condon h valus α α allow o achv h rurd accurac Fg Error racors - m Compur-add modlng dmonsrad good oprabl of h dsgnd ssms CONCLUSION apr consdrs h problm of dcnralzd conrol wh an nomnal modl for nrconncd ssm wh unnown paramrs and an unnown ordr whn drvavs of npu and oupu sgnals of h local subssms canno b masurd Consdrd robus conrol ssm allows compnsang paramrc and rnal dsurbancs wh gvn accurac δ for h prod of m alus δ and can b small nough usng h appropra paramrs of h closd loop ssm I s ncssar o no ha h closd loop ssm s funconng as an mplcl dfnd nomnal modl and paramrs of h modl ar usd n conrol algorhm I s mporan o no ha consdrd algorhm rmans h sam f hr s sa dla n a ssm as wll as n h cas whn a plan s saonar wh unnown paramrs whch valus ar lmd b a cran boundd s Bsds h advanag of h suggsd algorhm consss n h fac ha h srucur of a local conrollr s concdd wh h srucur of a local conrollr of sngl-conncd ssm hs gvs an advanag for h conrol of spaall dsrbud ssms h drawbac of h algorhm s a lac of an m analcall provd mhod of slcon of h paramrs of h conrollr EFEENCES [] B ola S Schrbaov obus sabl and conrol Naua [] O Nforov Adapv and robus conrol wh dsurbanc compnsaon S-rsburg Naua [] N Buov Ssm mbddng Analcal approach o analss and snhss of mar ssm ublshng hous of scnfc lraur of Bocharva NF 6 [] O Nforov Ernal dsurbancs obsrvrs Obcs wh nown paramrs Auom lmh no pp - [] O Nforov Ernal dsurbancs obsrvrs Obcs wh unnown paramrs Auom lmh no pp -8 [6] O Nforov Non-lnar conrol ssm wh drmnd dsurbancs compnsaon Izvsa Aadm Nau ora Ssm Upravlna 997 no pp 69-7 [7] I roshn O Nforov A L Fradov Nonlnar adapv conrol of compl dnamc ssms S rsburg Naua [8] N Buov N I Slvsu Analcal snhss of robus rgulaors basd on paramrcal uaons of Lur-a Auom lmh 7 no pp 6-6 [9] A A Bobsov Algorhm of robus oupu conrol of lnar obc wh compnsaon of unnown drmnd dsurbanc Izvsa Aadm Nau ora Ssm Upravlna no pp 9-97 [] A A Bobsov obus conrol algorhm of uncran obc whou masurng drvavs of adusd varabl Auom lmh no 8 pp 8-96 [] A sunov obus oupu conrol of lnar dnamc obcs harona avomazasa upravln 8 no 8 pp7- [] Gurs Analss and snhss of conrol ssm wh h dla oscow ashnosron 97 [] B Kolmanovs Nosov Sabl and prodc rgms wh dla oscow Naua 98 [] zvan Absolu sabl of auomac dla ssms oscow Naua 98 [] A sunov Adapv conrol of plans wh afrffc oscow Naua 98 [6] Yanushvs Conrol plans wh dlas oscow Naua 978 [7] G ao A Ioannou odl rfrnc adapv conrol for plans wh unnown rlav dgr IEEE rans Auoma Conrol 99 vol 8 no 6 pp 976-98 [8] J B oang DS Brnsn Drc adapv command followng and dsurbanc rcon for mnmum phas ssms wh unnown rlav dgr In J of Adapv Conrol and Sgnal rocssng 7 vol pp 9-7 [9] I B Fura A sunov obus conrol of unsad nonlnar obcs wh undfnd srucur roblms of Conrol 8 no pp -7 [] B rn an-su so Adapv dcnralzd conrol of dnamc ssms Bsh Ilm 99 [] A Brusn A class of sngular dsrbud adapv ssms Auom lmh 99 no pp9-7 9