Leader-Following Consensus of Nonlinear Multi-Agent Systems Based on Parameterized Lyapunov Function

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1 ODRES JOURL OF ELECRICL EGIEERIG VOL 5 O 2 SUER 25 3 Leade-Followg Cosesus of olea ul-ge Sysems Based o Paameezed Lyauov Fuco Pegah aba Saad 2 ohammad ehd ada okha Shasadegh Behouz Safaeada bsac hs ae sudes he cosesus oblem of olea leade-followg mul-age sysems (S). o do hs he eo dyamcs bewee he leade age ad followe oes ae descbed va a akag-sugeo (S) fuzzy model. If he obaed S fuzzy model s sable he all of he olea ages each cosesus. he cosesus oblem s vesgaed based o he aameezed o fuzzy Lyauov fuco combed wh a echque of oducg slack maces. he slack maces cause o decoule he Lyauov maces fom sysems oes ad heefoe suffce cosesus codos ae obaed ems of lea max equales (LIs). he oosed slack maces add a exa degee-of-feedom o he LI codos ad also decease he cosevaveess of he LI-based codos. Fally ode o llusae he effecveess ad mes of he oosed mehod a umecal examle fo he cosesus oblem of olea leade-followe S wh hee followes s solved. Idex ems olea mul-age sysems Cosesus akag- Sugeo (-S) fuzzy model Paameezed Lyauov fuco Lea ax Iequaly (LI). F I. IRODUCIO uzzy model based (FB) cool ovdes a famewok o desg a olea cool saegy fo a geeal class of olea sysems []. he hyscal heomea ae heely olea. I ode o ovecome he comlexy of he sysem oleay akag ad Sugeo ( 985) eseed a sysemac mul-modelg aoach called akag-sugeo (S) fuzzy model. he S fuzzy model lays a ccal ole FB cool []. I S fuzzy model whch oduced va fuzzy IF-HE ules he smooh olea sysem s eeseed va some local subsysems. he by fuzzy bledg of he local subsysems he oveall fuzzy model wll be acheved a covex sucue. he S fuzzy model ca be cosuced va he defcao mehods based o he u-ouu daa o ca be deved fom he exsece smooh olea sysem equaos [2]. I ode o cool he olea sysem based o he S fuzzy model seveal fuzzy cool mehods have bee eseed such as aallel dsbued comesao (PDC) o o-aallel dsbued comesao (o-pdc) [3]. Fuhemoe suffce sably codos of he closed-loo sysem wll be obaed based o he Lyauov dec mehod ems of LIs [3]. Oe of he commo aoaches o fomulae sably aalyss of he S fuzzy sysems s Lyauov dec mehod. he suffce sably codos ae coveed o LIs ad solved va covex omzao echques [2]. I some suaos may be he S fuzzy sysem s sable bu he commo quadac Lyauov fuco (QLF) does o exs [4]. hus seveal kds of LFs ae eseed leaue such as aameezed o fuzzy LF (FLF) [5] ax- LF [6] ad ecewse LF [7]. he ecewse LF s acheved based o a combao of some seaae quadac LFs whee each of hese quadac LFs s vald a acula ego. hus he ecewse LF suffes fom he oblem of dscouy bouday os of each ego. he FLF s also kow as bassdeede LF ad o-quadac LF. he FLF s acheved based o fuzzy aggegao of some quadac LFs [4]. ul-age sysems (S) ae cosuced by mulle ecoecg of ellge subsysems called ages. he ages o ogehe o sudy he oblems ha ae usually vey dffcul o eve mossble fo each age o solve [8]. Cosesus of Ss has aaced los of aeos as a ew feld of eseach. he cosesus oblem s vesgaed vaous backgoud of eseach such as: cool oboc bology ad comue scece backgouds. Roughly seakg f he ages of S each a ageeme o a secfc ceo he he cosesus oblem s feasble [9]. ccodg o he cool egeeg o of vew he cosesus oblem s coollg of he ages such ha he cosesus codos ae sasfed. Cosesus of Ss s a omsg eseach oc dug ece yeas. Cool heoy lays a moa ole fo solvg a cosesus oblem. Seveal kds of coveoal cool oocol ae aled o S o solve he oblem of cosesus such as: H cool oocol [] Pg cool [] ad samled-daa cool [2]. Desgg cool oocol fo lea dscee-me ages o solve he cosesus oblem s cosdeed [3]. oeove he cosesus oblem s suded fo Ss ha he dyamc behavo of he ages ae lea [4 H 5] ad olea [6]. Ref. [6] sudes a leadefollowg cosesus oblem of olea Ss. he cosesus oblem s o aoachs he olea followe ages o he ufoced leade age. hus he olea eo dyamcs bewee he followe ages ad ufoced leade ausc eceved Jue 25 25; acceed Seembe Deame of Eleccal ad Elecocs Egeeg Shaz Uvesy of echology Shaz Ia. {.aba m.mada shasadegh

2 SDI e al LEDER-FOLLOWIG COSESUS OF OLIER ULI-GE SYSES 3 age ae acheved. Subsequely he S fuzzy model of olea eo dyamcs ae calculaed ad he suffce sably codos ae deved ems of LIs. s afoemeoed Ref. [4] vesgaes he oblem of sably aalyss of S fuzzy sysems based o FLF ad oducg some slack maces. Slack maces geeae a degee of feedom LI codos ad have a dec effec o coveg he sably codos o he LI oes [4]. Wheeas by ceasg he umbe of ages he dmeso of LI codos wll be ceased ad subsequely he feasbly of LI codos wll be coveed o a challegg oblem. ccodg o he auhos bes kowledge hs ae es he fs aem o aalyze he cosesus oblem of olea Ss wh FLF ad slack maces. I hs ae we seek o solve he cosesus cool oblem moe elax scheme by usg he FLF ad oducg slack maces. Fs defe he leade-followg as he cosesus oblem. Secod he olea eo dyamcs bewee he ages of olea ufoced leades ad he followes wll be acheved. hus he cosesus oblem wll be coveed o he sably aalyss oe. hd based o he olea eo dyamcs he exac S fuzzy model wll be calculaed va seco olea aoach. Fouh ode o aalyze he sably of he S fuzzy model based closed-loo sysem he FLF ad some ew ull ems wll be defed ad suffce sably codos wll be deved ems of LIs. he ma cobuos of hs ae ca be classfed as follows:. he FLF wll be used o solve he cosesus oblem. 2. Some ew ull ems wll be defed. Slack maces ull ems cease he degee of feedom ad also coveed he sably codos o he LI oes. 3. he cool oocol wll be desged. 4. Comae o he ece ublshed woks [6 8-2] he LIbased sably codos ae moe elaxed. Fally he oosed aoach s aled o he cosesus oblem of olea leade-followe S wh hee followes. he emade of hs ae s ogazed as follows. Seco II s dvded o wo as. Fs a eses some basc coces of gah heoy ad secod a dscusses abou oblem fomulao. he ma esuls ae gve Seco III. I seco IV smulaos ae caed ou o llusae he effecveess of he ma esuls. Fally coclusos ae daw seco V. I he cue ae he suesc sads fo max asose deoes he -dmesoal Eucldea sace ad dag sads fo a block-dagoal max. I symmec block maces s used o eese a em ha s duced by I symmey s a dey max of dmeso ad deoes he Koecke oduc. 2. PRELIIRIES D PROBLE DESCRIPIO. Basc Coces o Gah heoy S cossg ages eeseed by a udeced gah G V G v v 2 v cosss of a veex se a edge v v se E G : E G v v v v V G meas ha age. If ca sed s fomao o he age ad vce vesa. I ohe wods hee s a deced coeco fom ode o ode. lso a adacecy max s defed such ha fo a deced coeco fom ode o ode f v v E G. Fuhemoe s suosed ha a he eghbo se of ode s deoed v V G : v v E G by. he Lalaca max assocaed wh G s defed as follows: ak l () k k a gah coag he leade ad all followes s eeseed by G. B. Poblem Fomulao Cosde a gou of followe ages ad oe leade. he dyamc of each followe age s gve by... x f x u (2) v a x f x whee s he sae of age (ode) s a olea couously dffeeable veco fuco eeseg he sc olea dyamcs of he -h age u ad s he cool oocol o be desged. I s assumed ha he leade has he followg olea mevayg dyamc s f s (3) s whee s he saes veco of he leade ad should be acked by all he followes. I ca be a equlbum o a chaoc ob o a eodc ob [6]. e ssumg ha deoes he eo bewee he saes of he followes ad he saes of he leade.e. e x s he eo dyamcs ca be eeseed as e f x f s u... Cosde a dsbued cosesus oocol as follows: (4)

3 ODRES JOURL OF ELECRICL EGIEERIG VOL 5 O 2 SUER 25 cd s () x ()... u c a x x (5) whee deoes he eghbog se of ode s he coulg segh R s he feedback ga max o be desged also should be osve defe [6]. eeses he coulg segh of he fomao flowg fom ode o ode ad a a whe access o he leade s sae fomao d d a c. If age has ad ohewse. Based o he S fuzzy model oduced [6] we ewe g x e f x f s g x e h x e as whee ca be aoxmaed by a S fuzzy model wh he chose emse vaables meawhle h x e cao be aoxmaed bu ca be esaed as x a oduc of bouded me-vayg max ad e. he fuzzy model of (4) s eeseed as he followg ules: Rule : If k s k he s ad 2 u e e x e s 2 ad ad 2 k whee s he emse 2... vaable veco fo ad l 2... k eeses he fuzzy ses s he umbe of f-he ules ad s he cosa max. he comac fom of he fuzzy model s as follows: whee l e h e x e u... h l l l l l whee l. he k (6) (7) s he gade of he membesh of l h h ssumo : x x Q osve defe max ad ca be we as 3. HE I RESULS whee Q Q R R. 32 (8) s a I hs seco based o he S fuzzy modelg he cosesus oblem of olea S s vesgaed. he eo dyamcs of (6) ca be we as follows: e ( h )e x e cl e whee I x I x 2 e e e... e L L dag d d 2... d o oba ew sably codos fo S cosde he followg ull max dees ha ae obaed fom (7) ad (8): (9) h h h e 3e () e e 2 e h cl e () has a lea elao wh 2 I s assumed ha 2 whee s a abay kow scala umbe 3 ad s a symmec max wh aoae dmeso. heoem : he followe ages as descbed (2) ca each cosesus wh he leade f G s coeced o equvalely a leas oe age each coeced comoe of G has access o he sae fomao of he leade ad fuhe hee exs symmec maces ad 3 ay max osve defe max ad scala such ha he followg LIs... hold fo. (2) P 3 P c I P (3)

4 SDI e al LEDER-FOLLOWIG COSESUS OF OLIER ULI-GE SYSES 33 H H 2 2 H I P.5 I (I R R) c L S H E (I ( I R R)) c L S whee E I P 3 I S (4) Poof: he followg aameezed Lyauov fuco caddae s chose V h e P e (5) V By dffeeag oe has h q V e Pq e h e Pe q q hq e I P q e he I P e (6) h If whee... 3 ad I P he V() h Z (2) e e H 3 H 4 H I P cl 3 2 H 4 E 2 cl V If he fo ay By usg assumo defg I emembeg he cosa H 5 H oe has H I P (.5 I (I R R) cl ) H E (I ( I R R)) ( cl ) ad (2) By usg he dees ()-() oe has hq V e I Pq e q q h e I P e hq e 3 e { e 2e e h cl e } q q q h h ( ( e I P e) h q e I P e e 3e q 2 e e e cl e Defe E I P 3 (7) (8) (9) o cove (2) o LI fuhe maulaos ae equed o do o he blea em ( cl ) of (2) as follows ( cl ) ( I ) ( cl ) cl Defe he ew decso vaable S oe has ( cl ) cl cl S Cosequely (2) s obaed as H H 2 2 H I P.5 I (I R R) c L S H E (I ( I R R)) c L S (22) heefoe he eos covege o zeo ad he cosesus s acheved. he oof s comleed. 4. ILLUSRIVE EXPLE I hs seco a umecal examle s gve o show he effecveess of he heoecal esuls.

5 ODRES JOURL OF ELECRICL EGIEERIG VOL 5 O 2 SUER 25 Examle: Cosde he mul-age ewok Fg. cossg of decal sysems [6]. he dyamcs of each ode s descbed by he followg chaoc equao x f x u (23) whee... x x 2 x x bx f x x 2x 2 x x 3 (24) 2 3 By choosg he values of 28 8 / 3 2 b fo chaos o emege he sysem (23) becomes he Loez sysem. he eo dyamcs sysem he dffeece bewee (23) ad he s f s leade e ca be deved as [7]: e e 2 e 2e 2 x e 3 e e 3 bx x e e e x e u x 2 3 (25) ek b k 2 ek x x 3 x 2 x x Remembeg assumo s bouded by Q R R whch 6 4 R. x e ca be calculaed accodg o (6) ad he membesh fucos ae cosdeed as [2]. Hee s suosed ha k d ad ohes ha meas oly age oe has access o he saes fomao of he leade. Solvg he LIs (-3) he ga max s obaed as follows: Fg. 2 shows he saes of he ages usg ad adomly chose al codos. I s cocluded fom Fg. 2 ha he cosesus ake lace quckly ad he followe ages ack he leade s saes fo he fuue. d k 34 fo Fg.. Coeced ewok oology. Hee we choose 3. he S fuzzy model (6) s used fo modelg he olea S as followg: Rule k : x e f s ad s he e ke x e u (26) k 2 whee x 2 e e e e 5 ad e 2 5 he augmeed eo sysem s as follows 4 e x e e x e u whee ek (27) 5. COCLUSIOS hs ae has cosdeed a cosesus cool oblem of leade-followg olea Ss. Ially he olea eo dyamcs bewee he leade age ad olea followe ages wee calculaed. hus he cosesus oblem of leadefollowg sysem chaged o he sably aalyss of he eo dyamc sysem. he he exac S fuzzy model of eo dyamc acheved va he coce of seco oleay. Suffce asymocally sably codos obaed based o he FLF ems of LIs. oeove based o he behavo of he closed-loo sysem some ew ull ems oosed. he slack maces defed ull ems had some advaages such as: ceasg he degee of feedom coveg he suffce sably codos o he LI oes decoulg he sysem maces fom he Lyauov oes ad also geeae moe elax codos. LI codos wee acheved by ulzg he Koecke oduc. he smulao esuls wee show he effecveess of he oosed aoach.

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