An Adaptive Fuzzy Control Method for Spacecrafts Based on T-S Model

Transcription

1 ELKOMNIKA, Vol., No., Noveber 20, pp. 6879~6888 e-issn: X 6879 An Aapve Fuzzy Conrol Meho for Spacecrafs Base on -S Moel Wang Q*, Gao an 2, He He School of Elecronc Inforaon Engneerng, X an echnologcal Unversy, X an 7002, P.R.Chna 2, School of Elecronc an Inforaon, Norhwesern Polyechncal Unversy, X an 70072, P.R.Chna *Corresponng auhor, e-al: wang@xau.eu.cn, gonlne@sna.co.cn 2 Absrac A oel reference aapve conrol eho s propose for unceran nonlnear characerscs of spacecraf aue conrol syse. hs eho cobnes fuzzy conrol ehoology an nonlnear feeback lnearzaon ehoology, whch ae he close-loop syse sable an he sae of fuzzy syse rack he sae of reference oel accorng o he parallel srbue copensaon heory an he raonal esgn of he fuzzy sae feeback conrol law. he nonlnear close-loop syse was lnearze by selecng fuzzy sae feeback paraeers an fuzzy ebershp funcon. hen an aapve conrol law was esgne by Lyapunov funcon. As a resul he syse can be aapve o all kns of paraeer unceranes an robus o oelng naccuracy an exernal surbance. Meanwhle, he sulaon resuls ncae ha he conrol law can uckly guaranee he sably of he spacecraf aue an be robus o oel perurbaons an exernal surbances. Keywors: spacecraf aue conrol, -S oel, oel reference aapve conrol, parallel srbue copensaon Copyrgh 20 Unversas Aha Dahlan. All rghs reserve.. Inroucon Hgher reureens have been pu forwar for he spacecraf aue conrol syse wh he ore coplcae srucure of a new generaon of spacecrafs an he ncreasng oern space ssons, such as he spacecraf aue capure, he reenry aue ajusen, spacecraf space ockng an so on. Because of he workng envronen an he srucure characerscs of he spacecraf, he nonlnear syse srucure, he unceran paraeers an so forh, he esgn of he spacecraf conrol syse suffers huge ffcules. herefore, reures ha he oern spacecraf conrol syse no only can fnsh varous space ssons, bu also have srong robusness, o ensure goo ynac characerscs an seay-sae ualy wh unceran paraeers, exernal surbances, srucure perurbaons an oher varous unceran facors effecs []. In he raonal nonlnear conrol eho, he conroller s esgne recly base on he non-lnear oel of he conrolle objec, by feeback lnearzaon an sophscae lnear syse heory. Fuzzy conrol eho base on -S oel s no recly base on he raonal non-lnear oel, bu base on he fuzzy oel bul by rules of he sae euaon. Copare wh he raonal nonlnear conrol, he bgges avanage of hs eho s he raonal conrol heory can be apple o he fuzzy syse sably analyss an conroller esgn. he conroller also can be esgne hrough he nroucon of unceran -S oel when he objec has boune uncerany. So he nonlnear conrol eho base on -S oel proves a new way for he oelng an conrol of coplex nonlnear ulvarable syse [2]. Fro he conrol effecs, he effecs of hese ehos epen on he approxaon egree of -S oel o he nonlnear objec, an he approxaon accuracy s prove a he cos of ncreasng he nuber of fuzzy local oels [-6]. In orer o reuce he nuber of fuzzy local oels whou affecng he conrol perforance of he syse, a seres of fuzzy aapve conrol algorhs wh robus perforance o he oelng errors an he uncerany has been researche [7-9]. In hs paper, a oel reference aapve conrol eho cobng fuzzy conrol ehoology an nonlnear feeback lnearzaon ehoology s propose for he spacecraf aue conrol syse wh unceran nonlnear characerscs, as -S oel escrbes. hs Receve Aprl 0, 20; Revse July 25, 20; Accepe Augus 7, 20

2 6880 e-issn: X eho esgne a raonal fuzzy sae feeback conrol law accorng o he parallel srbue copensaon heory, whch ae he close-loop syse sable an he sae of fuzzy syse rack he sae of reference oel. he nonlnear close-loop syse was lnearze by selecng fuzzy sae feeback paraeers an fuzzy ebershp funcon. hen accorng o he error euaon, an aapve conrol law was esgne by Lyapunov funcon eho. As a resul he syse can be robus o unceranes an oelng naccuracy. Meanwhle, he sulaon resuls ncae ha he conrol law can uckly guaranee he sable conrol on he aues of he arcraf, an be robus o oel perurbaons an exernal surbances. 2. Descrpon of he Spacecraf Aue Sably Conrol In orer o acheve hgh-precson aue conrol of he spacecraf wh unceran nonlnear characerscs, he -S oel s esablshe an spacecraf aue conrol law s esgne base on he gven spacecraf ynacs an kneacs euaon o realze hghprecson aue of he spacecraf conrol. 2.. Descrpon of he Spacecraf Aue Conrol Moel he ynac euaon of he rg spacecraf aue can be wren as [0-]: J J J () S( ) J Mc M Kneacs euaon base on Rorgues paraeers s: H ( ) (2) J for spacecraf nera arx, conrol orue vecor, R for surbance vecor, M S for skew syerc arx, an R for spacecraf angular velocy arx, H for hr-orer arx. M c R for R for Rorgues paraeer vecor, 0 2 S( ) () H( ) I S( ) 2 (4) can be consere as he vrual conrol npu of subsyse (2), so we esgn conrol law o sablze hs subsyse., k 0 (5) k Error vecor e s nrouce, e k (6) hus, he aue conrol syse () an (2) s rewren as: e J S( ek) J kh( ) e k J S( ek) J kh( ) J Mc J M (7) H ( ) e k H ( ) (8) ELKOMNIKA Vol., No., Noveber 20:

3 ELKOMNIKA e-issn: X 688 For he syse copose by (7)-(8), choose 0.5 k an copose e, 2,2,, 2,. Selec he sae uany,,,,, x e e e x, x, x, x, x, x, an efne e,,, x x, x, x x x x x , so he syse copose of (7) - (8) can be rewren as: x Axx ( ) Bu (9) In he forula, J S( xe 0.5 x) J 0.5 H( x) 0.5{ J S( xe 0.5 x) J 0.5 H( x )} Ax ( ) H( ) 0.5 H( ) J B 0, J M. 0, u Mc he conrol syse oel above s nonlnear oel. he corresponng -S oe us be esablshe o esgn he conroller by raonal conrol heory Descrpon of he Spacecraf Aue Conrol -S Moel In orer o ransfor he ynac oel of spacecraf aue no -S oel, can be ve no nne operang pons. xe, x [0,0],[0,2( 2)],[2,0],[2,2( 2)],[2,0],[ 2, 2( 2)], 2, A x Fgure. Mebershp Funcon Curve he followng rules can be rawn, Rule : If x s F (close o 0), x 2 s F 2 (close o 0), x s F (close o 0), x4 s F 4 (close o 0), x 5 s F 5 (close o 0), an x 6 s F 6 (close o 0), x Ax Bu. Rule 9: If x s F 9 (close o -2), x 2 s F 92 (close o -2), x s F 9 (close o -2), x 4 s F94 (close o -2), x5 s F 95 (close o -2), an x 6 s F 96 (close o -2), x Ax 9 Bu 9. Conserng he unceranes n he syse an he exernal surbances, he spacecraf aue ynacs -S oel can be escrbe as follows: R : f z s F an z s F p p hen x() A A x() B B u() () =,2,,, (0) In he forula, n conrol npu varable, x [ x,, x ] R n s he syse sae varable, u R s he z [ z,, z ] R p s he fuzzy aneceen varable. p An Aapve Fuzzy Conrol Meho for Spacecrafs base on -S Moel (Wang Q)

4 6882 e-issn: X s he fuzzy se, s he nuber of fuzzy rules,, A, B Fj,, ; j,, p A B s a arx wh he corresponng enson of he -h syse, s a srucure uncerany arx wh he corresponng enson of he -h syse, an () s he syse oelng errors an exernal nerferences. A, B an () are unknown. Usng he sngleon fuzzfcaon, prouc nference an he weghe average efuzzfcaon, wh he gven x, u, he oupu of he fuzzy syse s he weghe average of all he oupus n each subsyse. ha s, x () a ( z)[( A A) x( ) ( B B ) u( ) ( )] a ( z) ( z)[( A A) x( ) ( B B ) u( )] ( ) () p z Fj z j, Fj zj represen ha zj belongs o he ebershp of he j fuzzy se. F j, an z s he ebershp funcon. z z, an z. z z 0, so 0 z. Assung ha he arx A, B, 2,, s conrolle, naely syse (0) wh -S oel s locally conrollable, so he arx A, B can be ransfore o conrollable canoncal for by lnear sae ransforaon. herefore, whou loss of generaly, he arx A, B s se as follows. an an a2 a b () A 0, B 0, (). (2) Assung n he forula (2), he sybol of b reans unchange, ha s o say b s a posve or negave consan. Assung () represens boune oelng errors an exernal nerferences, naely () ax. hereby he spacecraf aue conrol -S oel s esablshe an he aapve fuzzy conroller s esgne.. Moel Reference Aapve Desgn base on -S Moel.. Moel Reference Feeback Conrol Law Desgn Accorng o he heory of parallel srbue copensaon, fuzzy sae feeback conroller s esgne by fuzzy conrol law as follows: () R : f z s G an z p s Gp hen u K x l r, 2,, () Gj,, ; j,, p s he fuzzy se, s he nuber of fuzzy rules, K s he sae feeback gan, K [ kn, kn,, k]. he whole sae feeback conrol law s: ELKOMNIKA Vol., No., Noveber 20:

5 ELKOMNIKA e-issn: X 688 z [ K x l r( )] u z Kxlr z [ ( )] (4) p z Gj z j, Gj z j represen zj belongs o he ebershp of he j fuzzy se G j, an z s he ebershp funcon. z z,an z. z z 0,so z (). 0. he close-loop syse can be obane by subsung Euaon (4) no Euaon x ( z) j( z){[( A A) ( B B) K j] x ( B B) ljr( )} ( ) (5) j Ignorng paraeer unceranes an exernal nerferences, naely A 0, B 0, () 0, Euaon (5) can be splfe as: x ( z) j( z)[( A BK j) x Bl jr( )] (6) j he followng lea s gven whou proof. Lea: If he fuzzy sae feeback conrol law s aken as (4) an (7) o (9), he fuzzy close-loop syse (6) can be ransfore no lnear syse (20). ( ks as a s ) s,2,, n (7) b l b, 2,, (8) b p G ( z ) b F ( z ), 2,, j, 2,, p (9) j j j j x () A x() B r() (20) A an an a2 a 0 0 0, B b 0. 0 hs lea shows ha when he syse oes no exs paraeer unceranes, exernal surbances an all he paraeers are precsely known, he saes () x of close-loop fuzzy syse (6) can rack he saes of lnear syse (20) hrough fuzzy sae feeback conrol law (4). As he reference oel, he lnear syse (20) s globally asypocally sable wh esre ynac perforance. hen he saes of close-loop fuzzy syse (6) can copleely rack he saes of syse (20). An Aapve Fuzzy Conrol Meho for Spacecrafs base on -S Moel (Wang Q)

6 6884 e-issn: X Fro he feeback conrol law (4) an (7)-(9), he values of A an B n forula (2) us be obane accuraely, ha s, he accurae values of aj, b (, 2,, ; j, 2,, n), n orer o rack he reference oel accuraely. However, for praccal syses, here are varous unceran paraeers A, B nse an exernal nerference (), an hese paraeers vary wh e, so we us esae he paraeers onlne by he aapve algorh..2. Fuzzy Aapve Law Desgn Because he paraeers A an B n he syse can no be obane accuraely ue o he srucural uncerany, he values of A an B us be esae on-lne by he aapve algorh, an hen replace he coeffcens n he fuzzy sae feeback conrol law wh he esae paraeers. In aon, here also exs oelng errors an exernal nerferences (), so a conrol law us be esgne o copensae he nerferences. hen we ge he cobnaon conrol law: u u u (2) f s par. In he forula, u s he sae feeback conrol par an u f s s he aapve conrol u f ˆ z [ Kˆ x lˆ r( )] ˆ z z ˆ us ax sgn[ e ( ) P ] z bˆ (22) (2) In he forula, ˆ ( z), Kˆ, lˆ are he esae values of ( z), K, l respecvely. By he forulas (7)-(9), A ˆ an B ˆ can be esae hrough he unknown paraeers oban he esae values of ( z), K, l. A an B o ˆ K [ ˆ, ˆ ˆ an an an an,, a a ] ˆ, 2,, (24) b ˆ l b, 2,, (25) ˆ b ˆ ( ) ˆ z b,2,, (26) Fro forula (), x za x B u z () ELKOMNIKA Vol., No., Noveber 20:

7 ELKOMNIKA e-issn: X 6885 z( A A ) x B u A x z () (27) Consruc he esae value xˆ( ) of x() o sasfy: x ˆ A xˆ z( Aˆ A ) x Bˆ u f z (28) Subsue he forula (22) an (24) o (26) no he above forula. xˆ A xˆ B r() (29) Copare he forula (29) wh he reference oel. xˆ x (0) hus, he esae error e x xˆ. e x x 2 s he sae as he rackng error he forula (27) subracs forula (28) o ge he unfe error euaon. ˆ z A x Bu z B e A e us () () z z A ˆ A A, ˆ B B B represen he esae paraeer errors. Forula () can be rewren as follows. e A e z z [ a,0,,0] x z z [ b,0,,0] u ˆ z b In us () z (2) In he forula, a [ an, an,, a], a ˆ a a, b ˆ b b. ˆ, ˆ a b n he esae paraeers A ˆ an B ˆ can be obane by he aapve law as follows. j j z () aˆ x() e () P z An Aapve Fuzzy Conrol Meho for Spacecrafs base on -S Moel (Wang Q)

8 6886 e-issn: X z (4) bˆ 2 e () Pu () z, 2 are he posve consans. P [ P, P2,, P n ] s a posve efne syerc arx an sasfes he soluon of Lyapunov euaon PA AP Q (Q s a posve efne syerc arx). Because he reference oel s asypocally sable, he soluon of Lyapunov euaon us exs. Also, because he oelng errors an he upper boun ax of exernal surbances () are ffcul o oban accuraely for he acual syse, as changes wh he runnng sae of he syse, he esae value of ax can be go by he aapve algorh of forula (5). ˆ e () P ax (5) An ax ax ax ˆ s efne as he esae error of he upper boun ax... Proof of he Conrol Syse Sably For he nonlnear unceran syse (0) base on -S oel, f he unknown paraeers are ajuse by aapve law (), (4) an (5) accorng o he oel reference aapve conrol law of (2), he close-loop syse s globally asypocally sable. he proof s he followng. Selec he uas-lyapunov euaon. a a b V() e () Pe() (6) 2 2 ax 2 Dervave of V () can be obane along he rajecory of he error syse (2). a a bb axax 2 V () e () Pe () e() Pe () (7) Subsue error euaon (2), an we oban: () () z a x e P z be Pu V () e Qe() 2 2 z z ˆ z b a ˆ a bb ˆ ˆ 2 e ( ) P u ( ) ax ax s 2 z (8) hen subsue he copensaon conrol (2). () () z a x e P z be Pu V () e Qe() 2 2 z a aˆ bb ˆ ˆ 2 e () P 2 e () P () ˆ ax ax ax 2 z (9) ELKOMNIKA Vol., No., Noveber 20:

9 ELKOMNIKA e-issn: X 6887 Because () ax, we can oban: ˆ z x e () P a V () e Qe() 2 a z () ˆ ˆ z e Pu b ax 2 b 2 ax e ( ) P 2 z (40) Subsue he aapve law (), (4) an (5). V () e Qe () 0 (4) When, he syseac error e () 0. So he close-loop fuzzy syse s asypocally sable. QED. 4. Sulaon Resul he spacecraf aue conrol law can be obane by he eho of fuzzy aapve conrol law base on -S oel, an he effecveness of he conrol law can be verfe by he nonlnear oel of he spacecraf as a conrolle objec. Dgal sulaon s ae by bulng SIMULINK conrol block agra n MALAB. he an oen of nera J x, Jy, J z s respecvely 204, 044 an 989(kg2), an he an oen of nera perurbaon J, J, J are all 5sn kg2. Assue plus nerference s x y z 0.05sn ra/s2, an he nal conon of sulaon s , he sulaon resuls are shown n Fgure Fgure 2. Rorgues Paraeer Change Curve Fgure. Change Curve Fgure 4. e Change Curve An Aapve Fuzzy Conrol Meho for Spacecrafs base on -S Moel (Wang Q)

10 6888 e-issn: X Fro he sulaon resuls, he conrol law propose can uckly conrol he nonzero nal aue of spacecraf o he balance poson. A he sae e, ceran robusness has been shown o oel perurbaons an exernal surbances. 5. Concluson For unceran nonlnear characerscs of spacecraf aue conrol syse, accorng o he parallel srbue copensaon heory, fuzzy conrol ehoology an nonlnear feeback lnearzaon ehoology are cobne o realze he lnearzaon of he close-loop syse nonlnear feeback by approprae choces of he fuzzy sae feeback coeffcen an fuzzy ebershp funcon. On hs bass, cobne wh Lyapunov's funcon, an aapve law s esgne, such ha he close-loop syse s sable, an he fuzzy syse sae can rack he sae of he reference oel o realze aapve conrol. he eho can aap o all kns of paraeer uncerany, an has goo robusness o oelng errors an uncerany. he sulaon resuls show ha he esgne conrol law can realze he spacecraf aue conrol effecvely an has goo robusness o he uncerany, oel perurbaons an exernal surbances. I s fully shown ha he -S oel has a broa applcaon prospec o hanle unceran nonlnear conrolle objec. References [] Zhang We, Guan Png, Lu Xao-He. Inrec Aapve Fuzzy H Conrol for Saelle Aue. Bejng Insue of Machnery, 20: 26(): [2] ong Shao-Cheng, Wang ao, Wang Yan-Png. Desgn of Fuzzy Syse an Sably Analyss. Bejng: Scence Press. 2004: [] Zeng Ke, Zhang Na-Yao, Xu Wen-L. Suffcen Conon for Lnear -S Fuzzy Syses as Unversal Approxaors. Aca Auoaca Snca. 200: 27(5): [4] Zeng Ke, Zhang Na-Yao, Xu Wen-L. ypcal -S Fuzzy Syses Are Unversal Approxaors. Conrol heory an Applcaons. 200: 8(2): [5] Cheng Chn-Hsng, Shu Sheng-l, Cheng Po-Jen. Aue Conrol of A Saelle Usng Fuzzy Conrollers. Exper Syses wh Applcaon. 2009: 6: [6] Juxang Dong, Youy Wang, Guang-Hong Yang. Conrol Synhess of Connuous-e -S Fuzzy Syses Wh Local Nonlnear Moels. Syses, Man, an Cybernecs, Par B: Cybernecs. IEEE ransacons on. 2009: 9: [7] Shyu Gao, Zhhong Man, Xnghuo Yu. Sablzaon of -S Ffuzzy Syses Usng LV Syse heory. Inellgen Conrol. IEEE Inernaonal Syposu on. 200: [8] Jeja LI, Ru QU, Yang CHEN. Consrucon Eupen Conrol Research Base on Precve echnology. ELKOMNIKA Inonesan Journal of Elecrcal Engneerng. 202: 0(5): [9] Jnhu Zhao, Yu Zhou, Langxun Shuo. A Suaon Awareness Moel of Syse Survvably Base on Varable Fuzzy Se. elkonka Inonesan Journal of Elecrcal Engneerng. 202: 0(8): [0] Zhang Hua-Guang, He X-Qn. Fuzzy Aapve Conrol heory an Applcaons. Bejng: Bejng Unversy of Aeronaucs an Asronaucs Press. 2002: -0. [] Huang Zhen-Gu, Zhao Zh-Jan. Large Spacecraf Dynacs an Conrol. Changsha: Naonal Unversy of Defense echnology Press. 990: -5. ELKOMNIKA Vol., No., Noveber 20: