ELEC 6041 LECTURE NOTES WEEK 3 Dr. Amir G. Aghdam Concordia University
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1 ecre Noe Prepared b r G. ghda EE 64 ETUE NTE WEE r. r G. ghda ocorda Uer eceraled orol e - Whe corol heor appled o a e ha co of geographcall eparaed copoe or a e cog of a large ber of p-op ao ofe dered o hae oe for of deceralao. each ao he coroller obere ol local e op ad corol ol local p. - oder a T e wh local corol ao ge b [ ] where he ae ecor ad r are he p ad op repecel of he h corol ao. The arce r ad j r j j are real coa arce. The e. ofe wre he followg for j j j - The e of local dac T feedback coroller for. are ge b Q...
2 ecre Noe Prepared b r G. ghda where η he ae ecor of he h feedback coroller he h local eeral p. The arce η η η r Q η ad r are real coa arce. The coroller. ca be wre he followg for Q where Q ad are block dagoal arce a follow Q Q Q ad ad are ge b. - Noe ha he -doa he coroller.4 wll hae he followg for V Y Y G G U U - Ug he ageed ae ecor oe ca wre he eqao for he cloed-loop e a follow For a rcl proper e he arce ad are ge b [ ] Q
3 where [ ]. - Noe ha he eqao for a ceraled coroller are alo lar o.4. Howeer for a deceraled coroller he arce Q ad are block dagoal a ge b.5 wherea he ceraled cae here o ch rerco he rcre of hee arce. - eceraled fed ode F [9] [] oder he -p r-op e. where r cora defed a follow { r r ad ae ha he deceraled flow r de } The a deceraled fed ode F of. wh repec o f p where p deoe he e of egeale of. oher word a F of. wh repec o f rak - For rcl proper e eqao.9 ca be plfed a p. - Th fac eqale o he followg p. r -. ad. ca be ed o fd he F of a e ercall. - Noe ha he e of F of. a be of he e of F of.... ecre Noe Prepared b r G. ghda
4 4 - arace of F [] oder he e. wh he deceraled dac coroller.4. ode p a F of e. ff for all T coroller of he for.4 a egeale of he cloed-loop e ar of.6. - The followg ercal algorh ca be ed o deere F of. Fd he egeale of. elec a arbrar block dagoal feedback ga ar ch ha he ar oglar. Th ca be accoplhed b e of a pedorado ber geeraor ercall beer o properl cale he ga ar ch ha. Fd he egeale of he ar c. 4 For alo all ereco e p p. he e of F wh repec o eqal o he c - Theore. [9] [] oder he e ge b. wh ad defed.7. e be he e of block dagoal arce defed.8. The a ecear ad ffce codo for he eece of a deceraled T coroller ge b.4 ch ha he cloed-loop e apocall able ha he e ha o F he cloed rgh-half cople plae. - Eaple. oder he e. wh ad he followg paraeer α [ ] [..]. ca be eal ee ha h e corollable ad oberable ad o ha o F. Howeer we hae ecre Noe Prepared b r G. ghda
5 5 ecre Noe Prepared b r G. ghda.... α e ca erf ha for. α h e ha a F a. wh repec o he dagoal forao flow. - The followg aalcal ehod ca be ed o deere F of. - Theore. [] oder he e ge b.. The p a deceraled fed ode of. wh repec o ff a oe of he followg codo hold rak ] rak[ rak for oe } { k k ch ha } { } {. 4 rak for oe } { k k ch ha } { } {. rak
6 6 for oe { } k ch ha } { }. k { eark. o be oed ha he codo ep of Theore. fac e ff he e oberable. larl he codo ep of Theore. fac e ff he e corollable. oher word f ep or ep of Theore. are afed he correpodg F alo a F of he e.. - Eaple. oder he e ge Eaple.. We wa o e Theore. o fd he ale of α for whch he e ha a F. We wll check he rak of arce ge ep o Theore.. The ar correpodg o ep of Theore. The ar correpodg o ep of Theore. [ ] The arce correpodg o ep of Theore. 4. ca be erfed ha for all egeale of { α } he arce ad are fll-rak. oher word he e oberable ad corollable. Howeer he rak of he ar 4 for α. wll be le ha a follow. rak 4 rak... Fro Theore. wll be coclded ha he e ha a F a. for α.. ecre Noe Prepared b r G. ghda
7 7 eferece [9]. H. Wag ad E. J. ao he ablao of deceraled corol e EEE Tra. oa. or. ol. -8 o. 5 pp c. 97. [] E. J. ao ad T. N. hag eceraled ablao ad pole age for geeral proper e EEE Tra. oa. or. ol. -5 o. 6 pp Je 99. []. F. Va ad E. J. ao he qaae characerao of approae deceraled fed ode g rao ero aheac of orol gal ad e ol. pp ecre Noe Prepared b r G. ghda
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