6.53 trl Systems trllaility ad Oservaility (hapter 6) Geeral Framewrk i State-Spae pprah Give a LTI system: x x u; y x (*) The system might e ustale r des t meet the required perfrmae spe. Hw a we imprve the situati? The mai apprah: Let u= v- Kx (state feedak) the x x (v - Kx); y x D(v - Kx) ( - K)x v; ( - DK)x - Dv The perfrmae f the system is haged y matrix K. Questis: Is there a matrix K s.t. -K is stale? a eig(-k) e mved t desired latis? These issues are related t the trllaility f (*)
Mai Result : The eigevalues f -K a e mved t ay desired latis iff the system (*) is trllale. ther situati: the state x is t mpletely availale. Oly a liear miati f x e.g. y = x a e measured. Hw a we realie u = v-kx? pssile sluti: uild a server t estimate x ased measuremet f y. Mai result : The server errr (differee etwee the real x ad estimated x) a e made aritrarily small withi aritrarily shrt time perid iff (*) is servale. We will arrive at these lusis i hapter 8. efre that we eed t prepare sme tls ad g thrugh these fudametal prlems: trllaility ad servaility. 3 trllaility: Defiiti sider the system x x u x R ; u R trllaility is a relatiship etwee state ad iput. Defiiti: The system r the pair () is said t e trllale if fr ay iitial state x()=x ad ay fial state x d there exist a fiite time T > ad a iput u(t) t[t] suh that T (T-τ) x(t) e x e u( τ)dτ x T mmet: There may exist differet T ad u that satisfy (). s a result there may e differet trajetries startig frm x ad ed at x d. trllaility des t are aut the differee. p. d () 4
t τ 'τ (tτ) '(tτ) W (t) e 'e dτ e 'e d Equivalet ditis: The fllwig are equivalet ditis fr the pair () t e trllale: ) W (t) is sigular fr every t >. ) W (t) is sigular fr at least e t >. 3) Fr every vr v v e t is t idetially er. 4) The matrix G = [ - ] has full rw rak i.e. (G ) =. 5) The matrix M() = [I ] has full rw rak at all. 6) M() has full rw rak at every eigevalues f. Nte: M() has full rw rak if is t a eigevalue f. We ly eed t hek the rak f M() at eigevalues f. t Nte : Of all the ditis ly 4) ad 6) a e pratially verified. 5 Example: Determie the trllaility fr x x u pprah : G [ ] a ( G) fr all pssile a - a - a ad The system t trllale whatever a ad are. pprah : hek M()=[-I ] at =- M( ) a (M(-)) < fr all pssile a ad Same lusi trllaility 6 3
Example: x x u a pprah : G [ ] a - a - ρ(g ) if a ad detg a ρ(g ) if either a r The system is trllale if a ad. pprah : hek M()=[-I ] at =- M( ) a M( ) a - (M(-))= (M(-))= iff a ad Same lusi trllaility 7 geeral SI system (diagaliale) x x u The ave system is trllale if ad ly if the eigevalues are distit ad e f the i s is er 8 4
Example: x x u G [ ] α β β α ρ(g ) if ad detg ( ) ρ(g ) if either r The system is trllale if ad ( ) () 9 Therem: sider the pair m m Suppse that the eigevalues f i ad thse f j are disjit fr i j. The () is trllale iff ( i i ) is trllale fr all i. 5
6 Therem: Let () = p. The pair () is trllale iff G -p+: = [ -p ] has full rw rak. This is equivalet t G -p+g -p+ eig sigular ad t G -p+g -p+ > (psitive defiite.) Example: 3 =4 p=. ()== p. p 4 G The first 4 lums are LI. (G -p+)=4 = () trllale
Example: G p 3 3 3 The first 3 lums are LI. The 4 th is depedet the first 3. 3 Hee () is trllale. 3 =4 p=. ()=. 8 4 9 9 3 8 has full rw rak 4 9 3 Effet f equivalee trasfrmati Reall that equivalee trasfrmati a make the struture leaer ad simplify aalysis. Questi: Des similarity trasfrmati retai the trllaility prperty? Therem: The trllaility prperty is ivariat uder ay equivalee trasfrmati Prf: sider () with G =[. - ]. Let the trasfrmati matrix e P. The () (PP - P) G [ ] - [P PP P P - [P P P ] - P[ ] PG - P P] Sie P is sigular ρ( G ) ρ(g ) 4 7
Next Prlem: Oservaility 5 Oservaility: dual ept sider a -dimesial p-iput q-utput system: x x u; u System y x Du ssume that we kw the iput ad a measure the utput ut has aess t the state. Defiiti: The system is said t e servale if fr ay ukw iitial state x() there exists a fiite t > suh that x() a e exatly evaluated ver [t ] frm the iput u ad the utput y. Otherwise the system is said t e uservale. y 6 8
Duality etwee trllaility ad servaility Therem f duality: The pair () is trllale if ad ly if ( ) = ( ) is servale. x x u Dual systems ' y ' 7 Equivalet ditis fr servaility: ) The pair () is servale. ) W (t) is sigular fr sme t >. 3) The servaility matrix G has full lum rak i.e. (G ) =. 4) The matrix M ( λ) λi has full lum rak at every eigevalue f. 8 9
9 Therem: The pair () is servale if ad ly if q -q G has full lum rak where q=(). Therem: The seraility prperty is ivariat uder ay equivalee trasfrmati; Therem: sider the pair m m Suppse that the eigevalues f i ad thse f j are disjit fr.i j. The () is servale iff ( i i ) is servale fr all i.
S far we have leared trllaility Oservaility Next we will study aial dempsiti: t divide the state spae it trllale/utrllale servale/uservale suspaes aial Dempsiti sider a LTI system x x u y x Du Let = Px where P is sigular the where x u y Du PP - P P D D Reall that uder a equivalee trasfrmati all prperties suh as staility trllaility ad servaility are preserved. We als have G PG G G P Next we are gig t use equivalee trasfrmati t tai ertai speifi strutures whih reflet trllaility ad servaility.
trllaility dempsiti Reall G =[ - ]. Suppse that (G ) = <. The G has at mst LI lums. They frm a asis fr the rage spae f G. Therem: Suppse that (G ) = <. Let Q e a sigular matrix whse first lums are LI lums f G. Let P=Q -. The p PP P R R Mrever the pair ( ) is trlla le ad ( si ) D (si ) D See page 59 fr the prf. 3 Disussi: fter state trasfrmati the equivalet system is u The iput u has effet. This part f state is utrllale. The first su-system is trllale if =. If the (t ) e ( ) e t t e (t-τ) u( τ)d Give a desired value fr say d. If we let t (t-τ) v( t) e e d '(tt) ad u(t) 'e W (t )[e t W ( t) t The yu a verify that (t )= d. e t (t-τ) e τ v( t ) ( τ)d 'e d ] ' d 4
3 5 Example: u x x =3 p= -p+=. Oly eed t hek G G ale utrll 3 ) ( G Q P q] [ Q Let q is piked t make Q sigular P PQ Nte: the last lum f Q is differet frm the k (page 6). s a result Ā is differet frm that i the k whih is. 6 Therem: Suppse that (G ) = <. Let P e a sigular matrix whse first rws are LI rws f G. The p R R R P PP q D ) (si D ) ( ad servale is ) ( pair the Mrever si Oservaility dempsiti (fllws frm duality) G Reall Disussi: fter state trasfrmati the equivalet system is Du y u u may e affeted y ut has effet y r
Summary fr tday: trllaility Oservaility aial dempsiti trllale/utrllale Oservale/uservale Next Time: trllaility ad servaility tiued trllaility/servaility dempsiti Miimal realiati ditis fr Jrda frm ditis Parallel results fr disrete-time systems trllaility after samplig State feedak desig (itrduti) 7 Prlem Set #9. Is the fllwig state equati trllale? servale? x x u y x If t trllale redue it t a trllale e; If t servale redue it t a servale e.. Is the fllwig state equati trllale? servale? x x u y x If t trllale redue it t a trllale e; If t servale redue it t a servale e. 8 4