Output Control of Nonlinear Systems with Unmodelled Dynamics 1

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1 Ppits of th 9th Wold Cogss h Itatioal Fdatio of Autoatic Cotol Cap ow South Afica. August -9 Output Cotol of Nolia Systs with Uodlld Dyaics Alxy A. Bobtsov *;** Sgy A. Kolyubi *** Ato A. Pyki * Alksad A. Kapitoov * Nikolay A. Nikolav * * Dpatt of Cotol Systs ad Ifoatics IMO Uivsity Kovkskiy p. 9 St. Ptsbug 97 RUSSIA ** Cotol of Coplx Systs Lab. Istitut fo Pobls of Mchaical Egiig RAS Bolshoy p. V.O. 6 St.Ptsbug 9978 RUSSIA *** Dpatt of Egiig Cybtics Nowgia Uivsity of Scic ad chology NO-79 odhi NORWAY -ail: bobtsov@ail.u s.kolyubi@gail.co a.pyki@gail.co. Abstact: I this pap w cosid th pobl of output cotol of olia systs i th psc of stuctual distubacs causd by so calld uodlld dyaics. his pap dvlops sults publishd i Bobtsov. I Bobtsov coditios of fficicy of coscutiv copsato w foud fo th cas of output stabilizatio of lia paatically uctai plat ud coditios of uodlld asyptotically stabl dyaics. W addd distubacs as sooth olia fuctio tig th coditios of scto stictio to th odl ad sythsizd gulato fo this cas. Kywods: sigula ptubatios olia systs fdback cotol stability cotol laws.. INRODUCION I this pap w cosid th pobl of output cotol of olia systs i th psc of stuctual distubacs causd by so calld paasit dyaics. A syst with uodlld dyaics ca b witt i th followig fo x f x t g x t z Гz du x t c z x R is stat vcto of th odl f x t ad g x t R a vcto fuctios u x t is a scala fuctio quatio dscibs uodlld dyaics is a sall costat paat z R is stat vcto of th odl. d ad c a costat atix ad vctos of cospodig disios. Wh tus ito zo th stuctu disio of th syst chags this kid of ptubatios is calld sigula ad th systs a calld sigulaly ptubd Fadkov t. al. 999 o slow-fast systs Bglug ad Gtz 6. ypical sigulaly ptubd systs iclud dict-div obots flxibl joit obots flxibl spac stuctus DC otos flxibl chaical systs tul diod cicuits aipla odl ivtd pdulu ad olia tiivaiat RLC twoks s fo istac Kokotovich t al 999 Naidu. R Nowadays th a ay paps ddicatd to pobls of aalysis ad cotol of th systs i coditios of stuctual sigula ptubatios s fo istac Kokotovich t al 999 Naidu Sai ad Lobi 6 Glig t. al. 978 Wag ad Sotag 6ab 7 8 Huag t. al. 9 Mastllo t. al. 7 Nguy ad Gajic Fadkov ad Adivsky ab Fadkov 987 Duzhiia t al. 996 Glig t. al I th paps Wag ad Sotag 6a 6b 7 8 authos cosid pobls of stability of sigulaly ptubd ootoous systs. I th paps [Huag t. al. 9 Mastllo t. al 7 Nguy ad Gajic pobls of cotol of lia sigula ptubd systs a cosidd. I th paps Fadkov ad Adivsky ab th task of cotol of olia sigulaly ptubd systs is cosidd. hs systs ca b witt i th followig fo x f x x u 3 t x f x x u t u is th cotol actio vaiabls R x is th vcto of slow x R is th vcto of fast vaiabls ad f f a th vcto fuctios of appopiat disios. Dspit activ dvlopt of cotol of olia sigulaly ptubd systs pobls of output cotol of olia uctai plats ud coditios of ifluc of uodlld dyaics a still op. his wok was fiacially suppotd by Govt of Russia Fdatio GOSZADANIE /9 Gat 7-U Pojct.Z5.3.3 Copyight IFAC 3

2 9th IFAC Wold Cogss Cap ow South Afica. August -9 his pap is ddicatd to aalysis of fficicy of output cotol law thod of coscutiv copsato cosidd i Bobtsov ad Nikolav 5 ud coditios of uodlld asyptotically stabl dyaics. W will cosid a olia plat A d y c F qu v l y b R is stat vcto of th syst 5; R is stat vcto of th syst 6; y R is asud output; fuctio v R is ot asud; u R is cotol; A F b c d q ad l a atixs ad vctos of pop disios; w suppos as i Fadkov t. al. 999 that F l q ; quatio 6 dfis asyptotically stabl dyaics i.. atix F is Huwitz uodlld fo dsig of cotol law; ub dtis spos spd of th syst; y is a sooth olia fuctio tig th v coditios of scto stictio of th viw ub C is ukow. 5 6 y C y h pupos of this wok is to fid th coditios which su stability of a syst with cotoll coscutiv copsato publishd i Bobtsov ad Nikolav 5.. MAIN RESUL Lt us wit syst 5 6 accodig to Bobtsov i th iput-output fo a p y t b p v t g p y 7 d p v t c p u t 8 p d / dt is difftiatio opato; output vaiabl y y t is asud but its divativs a ot asud; b s b s... b s b c p d s... as a s s a a s... ds d s d s d d s... gs a polyoials with g s g s g g ukow paats; s is coplx vaiabl; ; tasf fuctio b s a s has lativ dg of ; polyoial b s is Huwitz ad cofficit b. Lt us choos th followig cotol law Bobtsov ad Nikolav 5 u k p k k... k k y ub k ad polyoial p of dg a slctd fo th tasf fuctio s b s H s to b stictly positivly al a s k s b s positiv paat is usd fo copsatio of oliaity y ub k ad cofficits k i a coutd fo th syst to b asyptotically stabl fo zo iput y t. As w povd i Bobtsov ad Nikolav 5 tchically alizabl algoith 9 sus xpotial stability of th syst 5 6 fo th cas i.. i th absc of uodlld dyaics. But fo aalytical coditios of th cotol law 9 applicability w ot cosidd i th pap Bobtsov ad Nikolav 5. So w hav to fid stictios o ubs ad fo which th syst 5 is xpotially stabl. Lt us ak so tasfoatios. Substitutig 9 ito 8 w obtai v c p c p k p k p d p d p k p yˆ k p y c p yˆ ad y yˆ. d p h fo 7 w hav b p g p y v y a p a p p b p g p k y y a p a p p b p g p k y a p k p b p a p k p b p p b p y. a p k p b p Now lt us wit SISO odl i th fo of MIMO odl x Ax k b g y by 3 y c x x R is stat of 3; A b ad c a atixs of tasfoatio fo SISO odl ito MIMO odl ad by vitu of Yakubovitch-Kala la w ca fid sytical positivly dfiit atix P satisfyig th two followig atix quatios 33

3 9th IFAC Wold Cogss Cap ow South Afica. August -9 A P PA Pb c 5 Q N N 8 Q 3 Q Q is a positivly dfiit atix. Lt us wit ad i vcto-atix fo dk y h 6 z Fz q yˆ l z 7 R ad z R a stat vctos of th odls ad 7; atix... k k k 3... Huwitz i vitu of calculatio of paats k is k i of th odl d h ; F q ad l a atixs of tasfoatio fo th SISO odl to th MIMO o ad followig Fadkov t. al. 999 w suppos that Fl q. Lt us cosid dviatio vctos ly z 8 hy. 9 Aft difftiatio of quatios 8 9 w obtai l y ly Fz q F ly q y ly F q y y l ˆ hy hy dk y h y dk h y hy y h 3 dk h ad Fl q. So w hav a syst of difftial quatios x Ax k b g y by y c x ly F q l 5 hy h Positivly dfiit atixs Lyapuov quatios F. 6 R R ad N N satisfy R RF Q 7 Q Q ad Q3 Q3 a positivly dfiit atixs. Coditios of fficicy of th cotol law 9 fo stabilizatio of th syst a giv by th followig tho. ho. Lt th cotol law 9 is usd fo stabilizatio of th syst. Lt ub k sus that th tasf fuctio s b s H s is stictly positivly al. a s k s b s Lt positiv ubs ad t th coditios Q k l Rl Rq k Rlc b k l x Q x C Rlc g x Pbb Px x Qx fo all x ad. l + Rlc b 9 c Ax С x Pg h fo all tig th iquality 3 Q h Nh 3 Nhc b C Nhc g k Nhc b Q 7 ad fo th syst 5 is xpotially stabl. Poof of th tho ca b foud i Appdix. Rak. Lt us otic that coditios of th tho a ot coflictig. If w pass fo iqualitis fo quadatic fos 9 3 ad 3 o to iqualitis fo igvalus of th cospodig atixs ad akig so sipl tasfoatios fo 9 ad 3 w obtai l i { Q } l k k l R l c b q R q C l R l c 3 k g ad fo 3 w hav i i { Q} C g P g 33 b P b c AA c { Q 3 h } h N h k c ] b Nh [ c b C c g. 3 Idd it is asy to s that if w ultiply 3 ad 33 by ad td to zo th th iqualitis 3 ad 33 a coct ad coditios of th tho a ot cotadictoy. 3

4 9th IFAC Wold Cogss Cap ow South Afica. August -9 At th sa ti aalysis of coditio 3 allows us to coclud that ub should b icasd to su xpotial stability of th syst EXAMPLE h dsigd algoiths ca b usd fo such sigulaly ptubd plats as pdulu systs sigl-lik obots div with DC oto. I ths cass tasits of th cut cicuit of th DC oto a cosidd as paasit uodlld dyaics Mastllo t. al. 7 Fadkov ad Adivsky Li ad Li 7 sai t. al. 6 Cao ad Hovakiya 7 Yag ad Zhag 9 Yu t. al. 9. Lt us cosid stabilizatio of ubalacd oto to show fficicy of copsato ud coditios of uodlld dyaics Adivsky t al.. W hav th followig quatio of ubalacd oto dyaics takig ito accout optiizd cut cicuit of a lctic div J k g si k I 35 M I I k u. 36 J a itia ot ad ccticity of oto; is dbalac ass; k is fictio cofficit of baig as i Adivsky t al. w suppos that k ; is agl of dbalac dflctio fo vtical axs; I is cut of oto cicuit; k a costuctio paats of th div; M k is ti costat of optiizd cut cicuit; u is cotol voltag. Lt us wit syst 3 3 i th fo of 37 a si a 3v 38 y 39 F qu v a g / J a 3 k M / J q k ad F. Fist lt us siulat th syst 37 3 fo q ad k 5 figu i.. w do ot tak ito accout paasit dyaics. Figu 3 shows siulatio sults fo ad q h w tak ito accout paasit dyaics.. CONCLUSIONS I this pap w aalyzd fficicy of cotol law 9 fo stabilizatio of th syst 5 6. Cotol thod coscutiv copsato publishd i Bobtsov was show to b usd succssfully fo stabilizatio of olia paatically ad fuctioally uctai plat ud coctss of coditios Ralizatio of th coditio 3 was substatiatd i th ak. It is woth to ot that coditios 9 3 o 3 33 a difficult to chck i pactic at last ud assuptio of full paatical uctaity of th plat. But fo so kow gio of paat chag ths valuatios squi pactical ss. W also wat to ot that this sult lik ay oths i scic ay bco auxiliay. Fo istac this aalysis ca b usd fo dsig of cotol laws fo paat uctai systs with ixactly giv lativ dg. Pactical xapl of ubalacd oto stabilizatio was cosidd to illustat fficicy of coscutiv copsato thod ud coditios of uodlld dyaics. Fig.. Siulatio sults fo u Siulatio sults figu 3 show that accout of uodlld dyaics duc quality of tasits spos spd ad oscillatio icas but th closd-loop syst ais stabl. As i Adivsky t al. w assu a 33 ad a 7 ad siulat syst fo 3. Rsults of coput siulatio fo u a pstd i figu. Lt us ow choos cotol law i th viw 9 u k p k y 3 by vitu of dg of polyoial p is qual to o p p. Fig.. Siulatio sults fo 35

5 9th IFAC Wold Cogss Cap ow South Afica. August -9 Fig. 3. Siulatio sults fo REFERENCES Adivsky B.R. Blkha I.I. Botsov Yu.A. Gavilov S.V. Kooplv V.A. Lavov B.P. Polyakov N.D. ochia O.P. Fadkov A.L. Shstakov V.M. Cotol of Mchatoic Vibatio Dvics. Sait Ptsbug Nauka i Rusia. Bglug N. ad Gtz B. 6. Nois-Iducd phoa i slow-fast dyaical systs: a saplpaths appoach. Spig-Vlag Lodo Liitd Pitd i th Uitd Stats of Aica. Bobtsov A.A.. Robust Output-Cotol fo a Lia Syst with Uctai Cofficits. Autoatio ad Rot Cotol Bobtsov A.A. ad Nikolav N.A. 5. Fadkov thobasd dsig of th cotol of olia systs with fuctioal ad paatic uctaitis. Autoatio ad Rot Cotol Cao C. ad Hovakiya N. 7. L adaptiv cotoll fo systs i th psc of uodlld actuato dyaics. Pocdigs of th 6th IEEE Cofc o Dcisio ad Cotol Nw Olas LA USA. Duzhiia M.V. Nikifoov V.O. Fadkov A.L. 996 Mthods of output cotol of olia plats. Autoatio ad Rot Cotol # 3-33 i Rusia. Fadkov A.L. 987 Dsig of adaptiv cotol systs of olia sigulaly ptubd plats. Autoatio ad Rot Cotol #6 - i Rusia. Fadkov A.L. ad Adivsky B.R. a Sigula ptubatios of systs cotolld by gy-spdgadit thod. Poc. 3 IEEE Cof. Dc. Cotol Fadkov A.L. ad Adivsky B.R. b. Spd-gadit cotol of gy i sigulaly ptubd systs. Pp. 6th IFAC Syposiu o Nolia Cotol Systs NOLCOS Stuttgat Gay Fadkov A.L. Mioshik I.V. ad Nikifoov V.O. 999 Nolia ad adaptiv cotol of coplx systs. Dodcht: Kluw Acadic Publishs. Glig A.H. Loov G.A. Yakubovitch V.A Stability of olia systs with osigula quilibiu. Physathgiz Moskow. Huag Y. Cai C. ad Zou Y. 9 Output fdback fiit fqucy H cotol fo SISO sigulaly ptubd systs. Poc. of th 7th Asia Cotol Cofc Hog Kog Chia. Kokotović P.V. Khalil H.K. ad O'Rilly J Sigula ptubatio thods i cotol: aalysis ad dsig. SIAM. Li. S. ad Li K. 7. Coposit fuzzy cotol of olia sigulaly ptubd systs. IEEE tasactios o fuzzy systs 5. Mastllo S. Stipaovi c D.M. ad Spog M.W. 7 Multi-agt foatio cotol ad tajctoy tackig via sigula ptubatio. 6th IEEE Itatioal Cofc o Cotol Applicatios Pat of IEEE Multicofc o Systs ad Cotol Sigapo. Naidu D.S.. Sigula ptubatios ad ti scals i cotol thoy ad applicatios: A ovviw. Dyaics of Cotiuous Disct ad Ipulsiv Systs Sis B: Applicatios & Algoiths Nguy. ad Gajic Z. Solvig th sigulaly ptubd atix difftial Riccati quatio: a Lyapuov quatio appoach. Aica Cotol CofcMaiott Watfot Baltio MD USA. Sai. ad Loby C. 6. Sigula ptubatio thods i cotol thoy. No stadad thods ad applicatios i athatics Pisa. sai N. Hsu S. Ch C. ad Chiag C. 6. Digital slidig od cotol fo Maglv otos. Poc. of th 6 Aica Cotol Cofc Miapolis Misota USA. Wag L. ad Sotag E.D. 8. Sigulaly ptubd ooto systs ad a applicatio to doubl phosphoylatio cycls. J. olia scic Wag L. ad Sotag E.D. 6a. A ak o sigula ptubatios of stogly ooto systs. Poc. IEEE Cof. Dcisio ad Cotol Sa Digo Wag L. ad Sotag E.D. 6b. Alost global covgc i sigula ptubatios of stogly ooto systs. I Coault C. ad Machad N. d. Positiv Systs 5-. Spig-Vlag Bli/Hidlbg. Wag L. ad Sotag E.D. 7. Futh sults o sigulaly ptubd ooto systs with a applicatio to doubl phosphoylatio cycls. Poc. 6th IEEE Cofc o Dcisio ad Cotol. Nw Olas LA USA. Yag C. ad Zhag Q. 9 Multiobjctiv cotol fo S fuzzy sigulaly ptubd systs. IEEE tasactios o fuzzy systs 7. Yu H. Zhag X. Lu G. ad Zhg Y. 9 O th odl-basd twokd cotol fo sigulaly ptubd systs with olia uctaitis. Joit 8th IEEE Cofc o Dcisio ad Cotol ad 8th Chis Cotol Cofc Shaghai P.R. Chia. Appdix A Poof of th tho. Cosid Lyapuov fuctio of th followig viw V x Px R N. A. 36

6 9th IFAC Wold Cogss Cap ow South Afica. August -9 Difftiatio of A. with spct to quatios 6 allows us to wit V x A P PA x k x Pbl x Pby x Pg y F R RF Rqh Rlc Ax k Rlc bl Rlc g y Rlc N N Nhc k Nhc bl by Ax + Nhc g y Nhc by A. it y c Ax k bl g y by was usd istad of y. Lt us substitut quatios 5 7 ad 8 ito A. ad tak ito accout th followig xpssios x Pg y C x Pg C y C x Pg y k x Pbl x Pbb Px k l Rq h Rqh Rlc Ax Rl c Ax Rlc bl k Rlc b k l k Nhc Ax c Ax Nh Nhc bl k Nhc b l k Nhc g y C Nhc g C y C Nhc g y Nhc by Nhc b y y Rlc g C Rlc g C y C Rlc g y Rlc by Rlc b y y C y. h fo divativ of Lyapuov fuctio A. w obtai V x x Q x Q3 y Pbb Px Q k l Rq h Rl c Ax k Rlc b k l + c Ax Nh k Nhc b l + C x Pg + C Nhc g Nhc b y + C Rlc g is a ub. Rlc b A.3 Lt ad th fo so sall ad lag th ca b foud a positivly dfiit atix Q Q such that x Q x x Pbb Px + c Ax С x Pg Lt us choos such a way that iquality 3 x Qx. A. Q h Nh Nhc b C Nhc g + k Nhc b Q A.5 holds. h fo iquality A.3 w obtai V x Qx k l Q Q Rq Rl k Rlc b k l C Rlc g Lt th ub b such that Q k l l + Rlc b. A.6 Rq k Rlc b k l C Rlc g Rl Rlc b l + Q. A.7 h iquality A.6 taks th fo V x Qx Q Q i { Q } x. A.8 Fo A.8 w coclud that syst 6 is xpotially stabl ad hc syst 5 is xpotially stabl which was to b povd. 37