ELEC 604 LECTURE NOTES WEEK Dr mr G ghdam Cocorda Uverst Itroducto - Large-scale sstems are the mult-ut mult-outut (MIMO) sstems cosstg of geograhcall searated comoets - Eamles of large-scale sstems clude large fleble sace structures (LFSS) commucato etworks traffc etworks ower etworks etc - Whe cotrol theor s aled to a large-scale sstem or a multvarable sstem wth a large umber of teractg subsstems (such as a robot wth several jots) t s ofte desred to have some form of decetralzato I fact for such sstems t s ot realstc to assume that all outut measuremets ca be trasmtted to ever local cotrol stato - eamle of a 2-ut 2-outut sstem wth cetralzed ad decetralzed cotrol s gve Fgure (a) ad (b) resectvel Sstem (2-ut 2- outu R (s) Y (s) Cotroller (2-ut 2- outu (a) Cotroller (-ut - outu Sstem (2-ut 2- outu R (s) Y (s) Cotroller 2 (-ut - outu (b) Fgure : (a) cetralzed cotrol sstem (b) decetralzed cotrol sstem
- Tcal decetralzed large-scale sstems have several local cotrol statos whch observe ol local oututs ad cotrol ol local uts ll the cotrollers are volved however cotrollg the overall sstem - I ths course we wll be mal focused o decetralzed large-scale sstems - Decetralzed cotrol theor was develoed the ast 40 ears b ma researchers the feld of cotrol sstems The ma cotrbutos were tall made b E J Davso ad S Wag followed b other researchers such as Sljak Özgüer Khargoekar etc Durg the frst 0 ears the classcal decetralzed cotrol theor was develoed ad the some advaced tocs such as decetralzed adatve cotrol decetralzed otmzato ad decetralzed tme-varg cotrol were vestgated - I rearg the lecture otes of ths course the class otes of the courses largescale sstems I ad large-scale sstems II taught b Dr E J Davso at the Uverst of Toroto ad some addtoal aers have bee used Furthermore some of the assgmet roblems of ths course are the roblems that were desged b Dr Davso for the above-metoed courses - Ma of the decetralzed cotrol methods are smlar to ther cetralzed couterarts owever there are some mortat dffereces too Both smlartes ad dffereces wll be addressed ths course - We wll start wth some of the mortat ssues cetralzed cotrol of multvarable sstems - The state-sace model of a LTI sstem wth ratoal trasfer fucto ca be wrtte as follows: & ( ( Bu( ( C( Du( () where t R ( ) s the state vector u t R m ( ) s the ut vector ad t R r ( ) s the outut vector R B m R C R r ad D R r m are costat matrces The state equato () reresets a m-ut r-outut LTI sstem 2
- Cotrollablt: The state of the sstem rereseted b () s sad to be cotrollable f there ests a ecewse cotuous ut u ( that ca trasfer a tal state t ) to a fal state t ) Cotrollablt refers to the ablt to ( 0 ( f cotrol the sstem wth a cotrol ut u ( whch s a fucto of the state vector - The codto of cotrollablt deeds ol o the matrces ad B If the state of a sstem wth the state equatos () s cotrollable we sa that the ar ( s cotrollable - The sstem rereseted b () s cotrollable f ad ol f the m cotrollablt matr [ B BL B] s full-rak e: rak([ B BL B]) - lteratvel t s well-kow that the ar ( s cotrollable f ad ol f: where rak([ I B]) K K are the egevalues of the matr The above cotrollablt test s kow as the Poov-Belovch-autus (PB) cotrollablt test - Observablt: The state of the sstem rereseted b () s sad to be observable f there ests a fte tme T such that the tal state t ) ca be determed from the outut hstor ( 0 ( t t t T gve the ut u ( 0 0 Observablt refers to the ablt to estmate the state vector - The codto of observablt deeds ol o the matrces ad C If the state of a sstem wth the state equatos () s observable we sa that the ar ( C ) s observable - The sstem rereseted b () s observable f ad ol f the r observablt matr C C M C s full-rak e: 3
C C rak M C - lteratvel t s well-kow that the ar ( C ) s observable f ad ol f: I rak C K where K are the egevalues of the matr The above observablt test s kow as the Poov-Belovch-autus (PB) obserablt test - ll the roots of the characterstc equato of the sstem rereseted b () ca be laced a desred locato the s-lae usg a damc feedback f ad ol f the sstem s cotrollable ad observable - Cetralzed fed modes (CFM): Cosder the sstem () For smlct ad wthout loss of geeralt assume that D 0 Ths sstem s rereseted b the trle ( C The set of cetralzed fed modes (CFM) of ( C deoted b Λ ( C s defed as follows: CFMs of ( C I K R m r s( BKC) where s( BKC) deotes the set of egevalues of ( BKC) - The CFMs of the sstem ( C ca umercall be obtaed as follows []: ) Determe s( ) { K } multlctes cluded 2) Select a arbtrar ga matr K so that BKC where deotes the sectral orm of a matr ad s equal to the mamum sgular value of the matr Ths ca be accomlshed b use of a seudoradom umber geerator ad roer scalg of the K matr 3) The set of CFM s gve b the tersecto of the set of the egevalues of ad the set of the egevalues of ( BKC) - I the SISO case the set of CFMs s equvalet to the tersecto of the set of egevalues of the sstem ad the set of zeros of the sstem 4
- Whe a egevalue of s ot a CFM but t s ver close to beg a CFM (e for all ga matrces K oe of the egevalues of ( BKC) s ver close to oe of the egevalues of ) t s called a aromate cetralzed fed mode - I the SISO case a aromate CFM s a ole whch s ver close to a zero of the sstem For eamle see the followg ole-zero cofguratos Im{s} s-lae Im{s} s-lae Re{s} Re{s} CFM romate CFM (a) (b) Fgure 2: (a) Pole-zero cofgurato of a SISO LTI sstem wth a CFM; (b) ole-zero cofgurato of a SISO LTI sstem wth a aromate CFM - The defto of a CFM ca be used to determe certa roertes of the sstem ( C as follows []: ) To determe f ( s cotrollable: The ar ( s cotrollable ff the trle ( I has o CFM 2) To determe f ( s stablzable: The ar ( s stablzable ff the trle ( I has o CFM the closed rght-half lae of the comle lae 3) To determe f ( C ) s observable: The ar ( C ) s observable ff the trle C I ) has o CFM ( 4) To determe f ( C ) s detectable: The ar ( C ) s detectable ff the trle C I ) has o CFM the closed rght-half lae of the comle lae ( 5
5) To determe f ( C s cotrollable ad observable: The sstem rereseted b ( C s cotrollable ad observable ff the trle ( C has o CFM I other words the mode s ether ucotrollable or uobservable (or both) ff t s a CFM of ( C 6) To determe f ( C s stablzable ad detectable: The sstem rereseted b ( C s stablzable ad detectable ff the trle ( C has o CFM the closed rght-half lae of the comle lae - Oe ca also use the defto of a CFM to fd the mmal realzato of ( C through the followg algorthm []: ) Fd the CFMs of ( C ad let them be deoted b K } 0 < < { 2 2) Use a ga matr K so that the cotrollable ad observable modes of ( C BKC are all dstct ad dsjot from the CFMs 3) Fd the egevalues of ( BKC) ad let them be deoted b K K } { 2 4) Fd the egevectors of ( BKC) for the egevalues 2 K ad let them be deoted b 2 K resectvel 5) Fd the egevectors of ~ ( BKC) for the egevalues 2 K where the suerscrt deotes the ermta oerator (trasose of the comle cojugate) ad ~ deotes the comle cojugate Let these egevectors be deoted b 2 K resectvel 6) Normalze the egevectors 2 K so that 2 K 7) The mmal realzato of ( C BKC has the order ad s gve b ( C where: 6
7 B B C C M K L 2 2 2 : ) dag( : ] [ : ad the mmal realzato of ) ( B C s gve b ) ( B BKC C - It s to be oted that f the cotrollable ad observable modes of ) ( B C are all dstct ad dsjot from the CFMs } { 2 K oe ca sk Ste 2 ad go to the et ste b usg a zero ga matr 0 K - Note also that f all egevalues of are dstct oe ca fd the vectors stes (5) ad (6) drectl b formg the matr ] [ L L ad takg the verse of ths matr It ca be verfed that the rows of the verse matr are fact the vectors e: M M L L ] [ Trasmsso zero: Cosder the sstem () The comle umber s defed to be a trasmsso zero of () f there est a tal codto 0) ( ad a ut sgal < 0 0 0 ) ( t t e u t u t 0 t for whch the outut s equal to zero for all 0 t - It ca be show that the comle umber s a trasmsso zero of () ff the matr D C B I s ot full-rak whch meas that:
I rak C B < m( r m) D (2) - The vectors ad u ca be foud as follows From (2) t ca be cocluded that there ests a ozero vector ( r ) such that: R I C ad u are obtaed b decomosg where R ad u R r B D u 0 to two vectors as follows: The roof s gve the aed - The set of trasmsso zeros of a sstem ma be emt cota a fte umber of smmetrc comle umbers or clude the whole comle lae - sstem wth a set of trasmsso zeros equal to the whole comle lae s called a degeerate sstem Note that eve a cotrollable ad observable sstem wth full rak matrces B ad C ca be degeerate Throughout ths course we wll assume that the sstem s o-degeerate - gh ga outut feedback theorem [2] [3]: Cosder the sstem () ad wthout loss of geeralt assume that D 0 Let K m r R be a arbtrar matr wth rak( K ) m( r m) The f r m the fte egevalues of ρbkc as ρ cocde wth the trasmsso zeros of ( C ; f r m the for almost all K the trasmsso zeros of ( C are cotaed the fte egevalues of ρbkc - Note that ths s a geeralzato of the result for SISO sstems whch t s well kow that for hgh ga outut feedback the fte oles of the closed-loo sstem aroach the zeros of the sstem - It ca be show from the defto of CFM that the CFMs of a o-degeerate square sstem ( m r ) are cotaed the trasmsso zeros of ( C [] 8
- It s to be oted that trasmsso zeros are sometmes referred to as varat zeros The mathematcal defto of trasmsso zeros ad varat zeros are dfferet but we usuall assume that the are equvalet - I MTLB the commad tzero ca be used to fd the trasmsso zeros of a LTI sstem wth ratoal trasfer fucto ed: - The equalt (2) mles that f s a trasmsso zero of () the there ests a ozero vector ( r ) such that: R I C B D 0 () Decomose to two vectors R ad u R r as follows: u ssume ow that the tal state () s follows: ( 0) ad the ut sgal s as From () we wll have: ue u( 0 t ( I) Bu C Du t 0 t < 0 0 t 0 (2) 0 O the other had takg the Lalace trasform of both sdes of () we wll have: &( ( Bu( sx ( s) (0) X ( s) BU ( s) Ths meas that: 9
( si ) X ( s) (0) Bu s [( s ) s [ s s [ ( si) s Bu ] Bu ] ( I) B usg (2) the above equato wll be smlfed as follows: ( si ) X ( s) ( si ) s Bu ] From the above equato we have X ( s) whch results : s Therefore from () we wll have: ( e t t 0 t t t ( C( Du( C e Du e ( C Du ) e 0 t 0 Ths meas that f s a trasmsso zero of () the there est a tal codto ( 0) ad a ut sgal u t ( ue 0 t such that the outut s equal to zero for all t 0 Referece: [] E J Davso W Gesg ad S Wag algorthm for obtag the mmal realzato of a lear tme-varat sstem ad determg f a sstem s stablzable-detectable IEEE Tras utomat Cotr vol C-23 048-054 Dec 978 [2] E J Davso ad S Wag algorthm for calculato of trasmsso zeros of the sstem ( C B D) usg hgh ga outut feedback IEEE Tras utomat Cotr vol C-23 738-74 ug 978 [3] E J Davso ad S Wag Proertes ad calculato of trasmsso zeros of lear multvarable sstems utomatca vol 0 643-658 974 0