H Consensus of nonlinear complex multi-agent systems using dynamic output feedback controller: An LMI approach

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1 H oeu of olea cople ult-aget yte ug dyac output feedback cotolle: LM appoach l abaha Mahd Sojood dvaced otol Syte Laboatoy School of Electcal ad opute Egeeg abat Modae Uvety eha a. (oepodg utho btact: h pape vetgate a ew ethod fo coeu a goup of olea cople ult-aget yte ug fed-ode o-fagle dyac output feedback cotolle va a LM appoach. he popoed chee decetaled the ee that each aget ele o the elatve output foato aog the adjacet aget. he coeu baed cotolle ae deged to e the effect of olea te of the aget a well a eteal dtubace. ovetg coeu poble to tablato of a equvalet augeted yte ug pope tafoato Lyapuov tablty theoe appled to obta ukow cotolle paaete ode to guaatee coeu ad ultaeouly acque codeed cotol objectve. Fally to deotate the effectvee of the popoed algoth ad copae wth la eale eeache a uecal eaple o a ult-aget yte cotg of gle lk fleble apulato caed out. Keywod: olea Syte yac Output Feedback oeu Mult-aget Syte Lea Mat equalte (LM. toducto oodato ad coopeato of cople ult-aget yte (MS whch copoed of ultple autooou aget wth utual teacto ha bee a teetg eeach topc the lat decade. Local coucato betwee dvdual aget the etwok ay eult ceta deable global behavo uch a foato cotol flockg leade followg ad coeu. o each a ageeet egadg the tate of all aget a etwoked ult-aget yte defed a coeu. Recetly coeu ha bee wdely tuded becaue of t otewothy applcato uch a coopeatve cotol of uaed aplae ad udewate vehcle flockg of oble obot eo etwok ad edevou []. he coeu poble ha bee vetgated fo dffeet vewpot cludg vaou aget dyac cotat o etwok topologe cotol tatege ad deg ethod. May apect of coeu algoth aalyed fo ult-aget etwoked yte [] cludg covegece ethod obute to topology chage ad coucato delay. [] a adaptve dtbuted coeu poble of cotuou te ult-aget yte wth geeal lea dyac fo both cae of leadele ad leade followg coeu wth te vayg coucato topology deged. oeu a etwok of aget wth lea o leaed geeal dyac o a coeu ego ug tatc tate feedback toduced []. [5] a coeu potocol a ult-aget yte copoed of detcal aget wth geeal lea dyac of ay ode peeted codeg tatc ad wtchg etwok topology. Suffcet codto fo coeu lea ecod ode ult-aget yte wth dected etwok topologe ae deved [6]. odto o coeu algoth ult-aget yte wth ecod ode dyac cotag olea te ude dected topologe have bee tuded [7]. [8] leade followg coeu poble of ecod-ode olea ult-aget yte vetgated va pg cotol algoth. Ug a adaptve dtbuted potocol codeg eghbog tate coeu poble fo geeal lea ult-aget yte ad olea aget atfyg Lpcht codto addeed [9]. [] coeu poble fo low ode ult-aget yte wth heet Lpcht olea dyac ude dected coucato topology codeed. delay-depedet dtbuted ethod tuded [] to ue coeu olea ecod ode ult-aget of 5

2 yte ude wtchg topology. evethele a yteatc pocedue whch eue coeu fo ult-aget yte wth geeal lea dyac ad a olea te whch atfe uffcet codto eetal. tatc feedback coeu potocol fo detcal etwoked yte wth lea hgh-ode dyac toduce [] baed o oly output eaueet of the eghbog aget. State coeu poble of lea ult-aget yte by ultaeou tablato techque popoed [] ug eghbo-baed tatc ad dyac output-feedback cotol algoth. othe tatc output feedback cotolle fo H coeu a cla of lea te-vayg ult-aget yte wth both g foato ad odel ucetate deged []. Howeve ot pactcal cae the leaty of ult-aget yte dyac cotadcted hece ethod whch guaatee coeu ult-aget yte codeg olea dyac ae equed. oeu tackg ecod-ode ult-aget yte wth bouded dtubace ug a olea dtbuted chee tuded [5]. he obut coeu tackg poble [6] coce ult-aget yte wth uceta Lu e-type olea dyac wth a fed etwok topology utlg a potocol baed o elatve tate foato aog the eghbog et. Howeve fo eekg coeu ult-aget yte a geeal lea dyac cotolle whch ue oly eauable yte tate pefeable. Utlg at equalty techque [7] a coeu algoth fo geeal lea ult-aget yte ug local tate foato popoed. he appoach baed o gaph Laplaca of the etwok ad educe the coeu poble to tablato of the augeted yte. h algoth epaded to deg dyac output feedback cotolle [8] howeve the fal at equalte ae blea theefoe teatve ad accuate ethod of olvg lea Mat equalte (M ae eeded. Moeove [9] vetgate the output coeu poble of hghode ult-aget yte wth eteal dtubace ad popoe a dtbuted potocol ug the eghbo output foato. oethele oe pedefed codto o cotolle deg pocedue ae equed to covet the tally obtaed M to a LM fo. o the bet of ou kowledge o eult avalable o the LM-baed obut fed-ode decetaled dyac output feedback coeu algoth deg fo ult-aget yte wth olea dyac. Futheoe the etece of ucetate cotolle pleetato ha alo bee take to accout to cove cotolle faglty. he ethod offe the au poble boud o the aget oleate ad ultaeouly atteuate dtubace wth a pecbed atteuato level va LM faewok. he et of th pape ogaed a follow: Secto povde tateet ad oe pelay backgoud. Secto the poble foulated the popoed cotolle deg ethod peeted ad the theoetcal eult ae povded the fo of a theoe ad a coollay. he cae tude ae dcued Secto. Fally oe cocluo eak ae gve Secto 5.. Stateet ad pelae Soe atheatcal otato ae defed hee to be ued thoughout th pape. We ue. to defe the Eucldea o. Let ad deote colu vecto wth all eleet to be oe ad eo epectvely. Moeove dcate the detty at whle J all-oe at of deo. Fo the koecke poduct the ybol ued ad fo the two atce of ad wth the ae deo the Hadaad poduct a at of the ae deo a the opead wth eleet gve by: ( (.( j j j yetc potve defte at M defed by M o the othe had peudo vee of a gve o-quae at how by. of 5

3 Gaph theoy wdely ued to odel the coucato topology of the etwok of ult-aget yte. heefoe oe eceay ad equed cocept egadg gaph theoy ae toduced. he coucato etwok of a ult-aget yte copoed of aget uually odeled ug a dected gaph G { V E } whee V {... } a et of tege whch the vete dcate the aget whle the edge et E V V. f the foato of the aget avalable fo the oe the the pa a eleet of E ad depcted by j gaph epeetato. accodgly equvalet to. a udected gaph. th th ( j E he et of eghbo of aget weghted adjacecy at of G whee Laplaca of G defed a L ( j ( j E deoted by { j V : ( j E} ad a whee: ad a f j ( j V o a j th j th ] [ a j a othewe. he dag (deg...deg deg j a j othe wod the Laplaca gaph : l a l a j j... j j j j ( K V equece of edge the fo of ( (...( wth ceate a path a udected gaph. f thee et a path betwee each two vetce the udected gaph coected. t ca be poved that eo alway a ple egevalue of the Laplaca at of a coected gaph ad t coepodg egevecto whle all oeo egevalue ae potve.. oble foulato ode a ult-aget yte cope ca be epeeted by: aget.he olea dyac of the th aget u h ( t ( t y... ( whee u y q ad (t ae aget tate cotol put output ad the eteal dtubace of the th aget whle ad ae kow cotat atce of appopate deo. he pa of ad ( ae aued to be cotollable ad obevable ( epectvely. he h ( t cota all poble oleate ad ucetate of the th aget whch a pecewe cotuou vecto fucto atfe the followg quadatc equalty t doa of cotuty []: h ( t h ( t H H ( whee a boudg paaete ad uppoed to be aed whle H a cotat at. hu the ete ult-aget yte ca be epeeted a: u h( t ( t y v ( of 5

4 whee... ] u ] [ tate cotol put ad output epectvely. [ u u... u ad y [ y y... y ] ae the global the global yte at dag(... the global put at ad the global output at ad h t [ h ( t h ( t ] the global olea vecto fucto. Let ( [ H H whee H... H ] h( t ae defed ( ad dag... } wth v. he t alway poble to fd atce H ad { v uch that []: h( t H H H H (5 v whee H dag ( H H H wth atce of ad: dag v. (6 v H he uffcet codto fo (5 gve the followg codto fo atce H ad [6]: a ( H H a ( H H (7 he objectve to deg a potocol uch that the ult-aget yte ( ayptotcally eache global coeu o the tate whch epeed by: l ( t j ( t j (8 t Fo th pupoe a decetaled dyac output feedback cotolle popoed baed o the adjacet output foato betwee the aget. Hece the deged cotolle ay be etve to eo the cotol paaete the appled cotolle uually ecoute dffculte egadg accuace. h ue copel the dege to adopt a pocedue ode to avod poble udeable o utable cloed loop pefoace at pleet tage. heefoe the followg decetaled o-fagle dyac output feedback cotolle peeted to guaatee coeu aog aget ult-aget yte. ( u ( ( ( L y L y q q whee [... ] L q L q. ddtoally ad cotolle ukow paaete of the fo: dag( dag( dag( dag( (9 ae the global he petubato ucetate cotolle paaete atfy the o bouded codto of the fo: ( he cloed loop augeted ult-aget yte copoed of equato ( ad (9 ca be theefoe epeeted by: ( of 5

5 ( L ( q h( t ( t ( L q Sce ou dea to educe coeu poble ( to a tablato poble t eceay to tate two eetal obtacle towad the cotolle paaete attaet ad coeu auace. Ft f a cotolle deged fo ( uch that the oveall cloed loop yte table the the tal tate oly covege to eo howeve copellg coepodg tate to the ae value a teded coeu poble t eceay to defe a tafoato a: ~ L L L ( he tefeece ( baed o the fact that ~ povde elatve tate dffeece of the ultaget yte accodg to gaph Laplaca at hece f covege to eo the the coeu defto (8 acheved aog the aget. oethele a the ecod poble Lyapuov theoe whch the ajo cteo fo tablty aaly eque a full ak at yte tatepace epeetato that ot atfed ( afte applyg the tafoato ( due to ak defcecy of L. heefoe eovg oe of the ow of L wll elate yte edudacy accodgly a educed full ak yte at acheved by odfyg (: L ˆ L ˆ L ˆ whee Lˆ ( the depedet ow of L. ode to obta tate pace epeetato wth epect to the followg two lea ae toduced baed o gaph Laplaca popete. Lea [5]: L L L q L L oof: q q q ~ L L L ( ( Lea [5]: L L L L L L L oof: Ug Lˆ ad ubequetly L ( the oly eag challege to atch atce deo the eultg yte at. heefoe ad wll eplace ad epectvely whee L Lˆ hece the alteato of to eceay to atfy Lˆ L. otcg all the above etoed codeato the cloed loop yte of ( tafo to (5 afte eplacg ad applyg Lea ad Lea : l whee l l l h l Lˆ ( Lˆ ( Lˆ ( ˆ ( h t L t h. Sepaatg to two poto of ceta ad uceta paaete a: l (5 l whee (6 5 of 5

6 Lˆ Lˆ Lˆ ˆ L plfe ou pocedue towad tatg a theoe whch gve the uffcet codto o degg a fed-ode dyac output feedback cotolle fo coeu of a olea ult-aget yte eug pefoace of the cloed-loop yte (5. H efto. H H cla: Fo ay gve [ ] ad at H a cla of pecewe-cotue fucto defed by: wth appopate deo H { h( t h h h H H the doa of cotuty} (7 heefoe f h H the h( t whch ple that the equlbu of yte (5. efto. Robut tablty wth vecto degee [ ] : Syte (5 wth bouded olea ucetaty gve (5 called obutly table wth vecto degee f the equlbu globally ayptotcally table fo all h( t H. efto ad ae the vecto epao of thoe [] ad ae uppoed to be utled hee. heoe : odeg augeted cloed-loop yte (5 ad gve atce ad Q decbed by R dag... ad Q dag q... q wth ad a pecbed yetc ( ( potve defte weghtg atce f potve defte at q the fo of dag ( S S p dag... (8 ad atce j (... W dag ( w ( Z dag... w f ad k fo R.. ad potve cala fo j.. 5 et uch that the followg ato poble becoe feable: e ˆ ~ Δ Q ubject to Lˆ Lˆ 5 Hˆ 5 ˆ (9 whee 6 of 5

7 Δ ~ Q dag wth S S ( R Q Hˆ dag( H... H ˆ dag( ˆ... ˆ ( Lˆ k... k Lˆ J f... f c Z W W v v the the o-fagle dyac output feedback cotolle paaete of: W Z p p f k p p f... k... p p f k ake the oveall cloed-loop yte (5 table wth the obute degee vecto ad ultaeouly guaatee pefoace. [ ] oof: he dea adopted to poof heoe coce the obut tablato aaly of the yte (5 wth a optato faewok. odeg dtubace atteuato pefoace de wth epect to the ubequet equalty ut be atfed []: ~ V ( l lql ( whee V ( l a potve quadatc Lyapuov caddate fucto ad Q ~ a defed ( weght both the tate of ult-aget yte ad the cotolle ug pedefed R ad Q atce. he Lyapuov quadatc fucto caddate ha the fo of: V ( l ll ( he accodg to (5 V ca be epaded a: V ( l l ( l H h h. l l l l l l l l Wth decopoto of toduced (6 the equato (5 ca be ewtte a: l V ( l whee Lˆ ( l l l l H Lˆ 5 l l ( l [ l 5 ] 5 ( ( ( (5 (6 ad Lˆ h( t. toducg the equato (6 eaaged: 5 7 of 5

8 8 of 5 ˆ ˆ L L V l (7 Subttutg (7 to ( the ubequet at equalty obtaed: ˆ ˆ ~ L L Q (8 ccodg to ( the followg o boud equalte wth epect to ad ae uppoed to be atfed:. (9 Futheoe accodg to (5 oe ca coclude:. ˆ ˆ 5 5 H H ( he the cobato of equato (9 ad ( wth epect to : ˆ ˆ ( ( H H ( Repeatedly ug S-pocedue ([] fo (8 ad ( eult : (

9 whee: ~ Hˆ Q Δ 5 Hˆ Lˆ Lˆ 5 ( hu f due to eveal ultplcato of vaable. o deal wth th poble we ft epad accodg to defed (8. whee Lˆ Lˆ the codto ( atfed. Howeve the at equato ( ot a LM Lˆ Lˆ Lˆ p... p Lˆ J p... p c. Lˆ ow by chagg vaable a: W Z k p ad f p ad ( ad ubequetly ( ae obtaed epectvely. Fally a chu copleet appled ad toducg ew vaable ˆ 5 coplete the poof. oollay : ode ope-loop ult-aget yte ( ad the cotolle (9 wthout ucetate.e. ad. Gve atce R ad Q decbed heoe f c c c potve defte at W dag( w... w f ad fo ato poble feable: e ˆ ~ Q ubject to c of the fo (8 ad atce Z dag... k (.. ad potve cala et uch that the followg Hˆ ˆ whee Q ~ Ĥ ad ˆ ae all defed ( the the output feedback cotolle paaete gve ( ake the oveall cloed-loop yte table wth the obute degee vecto [ ] ad ultaeouly guaatee H pefoace. ( 9 of 5

10 oof: he poof pocedue la to that of heoe ug [ ] l 5. Reak : the cae that the ode of dyac output feedback cotolle (9 et to be eo the oollay ad heoe ae educed to tatc output feedback cotolle deg. addto f all the tate ae avalable aug a dyac tatc feedback cotolle ca be deged by the pocedue oollay ad heoe.. Sulato th ecto to valdate the effectvee of the popoed chee the deged dyac output feedback cotolle heoe ad oollay ae appled fo a ult-aget yte cotg gle a lk apulato. Ug Matlab/Yalp package fo olvg deved LM ulato eult ae peeted ad copaed wth la eale eeache. ode a ult-aget yte cotg of thee gle lk apulato wth fleble jot actuated by oto a depcted Fgue. Fgue : gle lk apulato wth fleble jot he dyac of a gle lk apulato wth fleble jot ca be epeed by []: J ll gl ( l k( l J k( b k u J l l J l whee ad epeet the oto agula otato the oto eta the lk agula otato ad the lk eta epectvely. Moeove the pote a half lk legth k tooal pg cotat. Vcou fcto coeffcet deoted by b ad the aplfe ga. O the othe had the devato of agula otato lk ad oto ae defed a agula velocty of the fo: l l ug that oly the tate of the oto ae avalable the tate pace decpto of a gle lk apulato wtte accodg to oal value []: u h ( t y whee l the tate vecto ad: l l k (5 (6 (7 of 5

11 h ( t he elected etwok topology of the ult-aget yte gve by Fgue...( l (8 Fgue : etwok topology of the ult-aget yte Hece the Laplaca at coepodg to Fgue ca be wtte accodg to (. L. o olve coeu poble ug the popoed ethod heoe ad coollay the peece of Gaua dtubace wth eo ea ad ut vaace we aue ecod ode cotolle wth uceta ad ceta paaete a two poble ceao. Let Hˆ R 8 8 ad Q 6 fo both vetgated ethod ad boud. 5. heoe wth the followg paaete ucetate fo petubed dyac output feedback cotolle:.5. (t (t (5t co(t. (t co(t co( t (t. ( t. he obtaed cotolle paaete accodg to heoe ad oollay ae collected Eo! Refeece ouce ot foud. ad the agula otato ad agula velocty of each apulato depcted Fgue. able. otolle paaete fo the aget ug ( heoe ad ( oollay. aaete get get get (9 c ( ( c ( of 5

12 c c ( ( ( ( ( get get get get get get Moto gula Rotato (ad.5 Moto gula Velocty (ad/ec e (ec e (ec.5 Lk gula Rotato (ad.5.5 get get get Lk gula Velocty (ad/ec - - get get get e (ec e (ec Fgue : State coeu gle lk apulato wth fleble jot ug the popoed cotolle heoe (blue ad oollay (black. ccodg to Fgue tate covegece of gle a lk apulato ha o codeable ocllato ug dyac output feedback cotolle. Moeove coeu lghtly fate whe the cotolle ae aued to be ceta whle the cloed-loop ult-aget yte epoe ea elable wth the etece of ucetate cotolle ug heoe. ode to copae the cotolle pefoace wth la wok coeu of Lpcht olea ult-aget yte wth the tatc tate feedback coeu potocol tuded [5] opted. Sulato eult Fgue how coeu aog thee gle lk apulato ug [5] wth the ae dyac coucato topology ad tal codto. of 5

13 .5 Moto gula oto (ad get get get Moto gula Velocty (ad/ec - - get get get e (ec e (ec.5 Lk gula Rotato (ad get get get Lk gula Velocty (ad/ec - get get get e (ec e (ec Fgue : State coeu gle lk apulato wth fleble jot ug the popoed cotolle [5] he tajectoe of cotolle [5] ad the popoed ethod heoe ae depcted Fgue 5 becaue of the potace of cotol effot ad ocllato of actuato cotolle pleetato. ( a ( b.5.5 get get get 5 get get get otol Effot otol Effot e (ec e (ec Fgue 5: cotolle effot of the popoed cotolle theoe (a ad [5] (b addto to a vey hgh ovehoot the begg tace the ocllato cotolle output ca be ee Fgue 5 (b cotat to a ooth ad o-fluctuated cotolle effot deved by the dyac output feedback popoed heoe. Moeove to aalytcally copae taet cotolle' effot fo the popoed potocol [5] ad theoe SE E SE ad E ([6] ae utled a cotolle pefoace dce. of 5

14 able. otolle pefoace dce fo popoed ethod : heoe ad [5]. otol Method get SE E SE E get heoe opoed cotolle [5] get get get get get eug the eult able t obvou the cotolle pefoace by ug the popoed ethod heoe uch oe acceptable accodg to all pefoace dce eve wth the peece of cotolle ucetate ad eteal dtubace. 5. ocluo th pape a coeu ethod fo a goup of olea ult-aget yte ug fed-ode o-fagle dyac output feedback cotolle popoed. We educed the coeu poble to a equvalet ultaget yte whch uppoed to be tabled ad a appopate Lyapuov fucto adopted to deve ukow decetaled cotolle paaete va a LM optato. he obut tablty of cloed loop augeted yte acheved wth the peece of olea dyac of the aget. Moeove the cotolle deg pocedue foulated to toleate o bouded petubato paaete ad a H pefoace de utled to atteuate eteal dtubace. Fally to deotate the effectvee of the popoed algoth ad copae wth la eale eeache a uecal eaple o a ult-aget yte cotg of gle lk fleble apulato caed out. Refeece. Zhag X. Lu X.: Futhe Reult o oeu of Secod-Ode Mult-get Syte Wth Eogeou tubace. EEE a. cut Syt. Regul. ap (.. Olfat-Sabe R. Fa J.. Muay R.M.: oeu ad coopeato etwoked ult-aget yte. oc. EEE (7.. L Z. Re W. Lu X. Xe L.: tbuted coeu of lea ult-aget yte wth adaptve dyac potocol. utoatca (.. L Z. ua Z. he G.: yac coeu of lea ult-aget yte. E otol heoy ppl (. 5. Wag J. heg. Hu X.: oeu of ult-aget lea dyac yte. a J. otol. 55 (8. 6. L H. Lao X. og. Xao L.: Secod-ode coeu eekg dected etwok of ult-aget dyacal yte va geealed lea local teacto potocol. olea y. 7 6 (. 7. Yu W. he G. ao M. Kuth J.: Secod-Ode oeu fo Multaget Syte Wth ected opologe ad olea yac. EEE a. Syt. Ma ybe. at ybe (. 8. Sog Q. ao J. Yu W.: Secod-ode leade-followg coeu of olea ult-aget yte va pg cotol. Syt. otol Lett (. 9. L Z. Re W. Lu X. Fu M.: oeu of Mult-get Syte Wth Geeal Lea ad Lpcht olea yac Ug tbuted daptve otocol. EEE a. uto. otol (.. Lu K. Xe G. Re W. Wag L.: oeu fo ult-aget yte wth heet olea dyac ude dected topologe. Syt. otol Lett (.. he K. Wag J. Zhag Y. Lu Z.: Secod-ode coeu of olea ult-aget yte wth etcted wtchg topology ad te delay. olea y (.. Zhag. Wag X. Meg L.: oeu poble fo hgh-ode L yte: a decetaled tatc output feedback ethod. t. J. ov. oput. f. otol. 9 5 (.. Hu Y. L. La J.: ef pape - oeu of ult-aget yte: a ultaeou tablato appoach. E otol heoy ppl ( of 5

15 . Wag Z. g. og H. Shu H.: coeu cotol fo ult-aget yte wth g eaueet: he fte-hoo cae. Syt. otol Lett (. 5. Zhao L.-W. Hua.-.: Fte-te coeu tackg of ecod-ode ult-aget yte va ogula SM. olea y (. 6. Zhao Y. ua Z. We G. he G.: Robut coeu tackg of ult-aget yte wth uceta lu e-type o-lea dyac. E otol heoy ppl (. 7. Zha G. Okuo S. ae J. Kobayah.: oeu algoth fo ult-aget yte: a at equalty baed appoach. etwokg Seg ad otol 9. S 9. teatoal ofeece o. pp EEE (9. 8. Zha G. Okuo S. ae J. Kobayah.: ew coeu algoth fo ult-aget yte va decetaled dyac output feedback. J. tell. Robot. Syt. 6 9 (. 9. Lu Y. Ja Y.: oeu poble of hgh-ode ult-aget yte wth eteal dtubace: H aaly appoach. t. J. Robut olea otol (.. Stakovć S.S. Šljak..: Robut tablato of olea tecoected yte by decetaled dyac output feedback. Syt. otol Lett (9.. Schee. Gahet. hlal M.: Multobjectve output-feedback cotol va LM optato. EEE a uto otol (997.. oyd S..: Lea at equalte yte ad cotol theoy. Sa (99.. etew.m. Maque H.J. Zhao Q.: dect apled-data deg appoach fo obot tablato. eca otol ofeece 8. pp (8.. Raghava S. Hedck J.K.: Obeve deg fo a cla of olea yte. t. J. otol ( L Z. Lu X. Fu M. Xe L.: Global H_f coeu of ult-aget yte wth Lpcht o-lea dyac. E otol heoy ppl. 6 8 (. 6. She S.M.: Mode otol Syte heoy ad eg. Joh Wley & So ( of 5