State Observer Design

Transcription

1 Sa Obsrvr Dsgn A. Khak Sdgh Conrol Sysms Group Faculy of Elcrcal and Compur Engnrng K. N. Toos Unvrsy of Tchnology Fbruary Problm Formulaon A ky assumpon n gnvalu assgnmn and sablzng sysms usng Sa Fdback: Accss o all sa varabls for sa fdback. A Powr Plan Supr Har has up o 80 sa varabls! In many cass, assumpons mad abou accss o all sa varabls ar mpraccal du o h: Non physcal sa varabls Th numbr of sa varabls Cos of snsors Srvc and mannanc ssus Envronmnal nos Rlably Snsor placmn 2 1

2 A Ky Obsrvaon: Sa varabls Esmaon or Obsrvaon s crucal for sa fdback conrol. Obsrvr: A dynamcal sysm ha s oupu s sa varabls smas. Hsorcal rvw of Obsrvr Dsgn problm Wnr (1942) Kalman (1960) Kalman and Bucy (1961): Kalman Flr Lunbrgr (1963): Obsrvrs 3 Th frs vw: Ral Sas ha can no b masurd Inpu Ral Sysm Sysm Oupu Modl Modl Oupu Modl Sas Imporan Qusons? 4 2

3 Improvmns mad n h srucur o rsolv som of h rasd ssus: Ral Sas ha can no b masurd Inpu Ral Sysm Sysm Oupu Modl Modl Oupu Modl Sas Esmaon Error 5 Applcaons of Obsrvr and Opmal Obsrvrs or Kalman Flrs Esmaon of sa varabls for conrol Faul Dagnoss Bhavour Prdcon of Dynamcal Procsss Mssl Gudanc Fnancal Engnrng: Sock Bhavour, Inflaon ra, Powr Engnrng: Load prdcon, Gnral Sysm Engnrng Bo Mdcal Engnrng

4 Obsrvr Srucur and s Proprs ( ) x () = Ax + Bu() u ( ) = Kx( ) y() = Cx() Sysm wh un-masurabl sa varabls Sa Fdback Conrol Masurabl Oupus Obsrvr Dynamcal Sysm: u () y() Obsrvr Dynamcal Sysm () x Dynamcal Equaons? 7 Obsrvr Dynamcal Equaons: ( ) x () = Ax + Bu() + Ly() Slc hs marcs such ha h Obsrvaon Errors ar Asympocally drvd o zro () = x () x () ( ) ( ) () = x() x() = Ax + Bu() [ Ax + Bu() + Ly()] 8 4

5 () = Ax + Bu() [ Ax + Bu() + Ly()] ( ) ( ) () = Ax + Bu () Ax [ ()] Bu () LCx () ( ) ( ) () = A () + ( A LC A) x() + ( B B ) u() Slc hs marcs n ordr o zro h obsrvaon rror asympocally and ndpndn from npu and sa vcors: A = A LC B = B Obsrvr Dsgn Marx 9 ( ) x() = Ax + Bu() + Ly() ( ) x() = ( A LC) x + Bu() + Ly() u () x() = ( A LC) x ( ) + [ B L] y () or ( ) x () = Ax + Bu() + L[ y() Cx()] Obsrvr Dynamcal Equaons 10 5

6 Block Dagram of Obsrvr Dynamcal Sysm 11 () = A () Dynamcal Sably of Obsrvaon Error Sysm. Obsrvr Pols Trad-off n choosng h Pol loc Kalman Flr Sngl oupu and Mul-oupu sysm. Ncssary and Suffcn condons for h Obsrvr Dsgn? 12 6

7 Ncssary and Suffcn condons for Obsrvr Dsgn T T T d[ λi ( A LC)] = d[ λi ( A C L )] Dualy Prncpl: Th ncssary and Suffcn Condon for Obsrvr Dsgn s Sysm Obsrvably. 13 Sa Fdback Conrol Sysms Dsgn wh Obsrvr Sa fdback conrol sysm srucur wh a full ordr obsrvr: ( ) ( x( ) = Ax + Bu ) y( ) = Cx( ) u ( ) = Kx( ) ( ) x () = Ax + Bu() + L[ y() Cx()] u ( ) = K x ( ) ( ) x () = [ A BK LC] x + Ly() 14 7

8 Closd Loop Block Dagram: 15 Closd Loop Sysm Analyss: x () A BK x() = LC A LC BK x () x() () = x () x () () = ( A LC )() x () = ( A BK ) x () + BK() lm () = 0 x () = [ A BK] x () Marx Slcon: Obsrvr Dsgn + Conrol 16 8

9 Sparaon Prncpl: I 0 x () A BK x() T = = I I LC A LC BK x () x() 1 I 0 T = I I x () A BK BK x() = 0 A LC ( ) () si A + BK BK d 0 si A LC = + d( si A + BK)d( si A + LC) Sparaon Prncpl 17 Dscr-Tm Obsrvrs Consdr h sa fdback dscr m conrol sysm: x( k + 1) = Gx( k) + Hu( k) y( k) = Cx( k) u( k) = Kx ~ ( k) Th closd loop dynamcal quaon: Prdcon Obsrvr x~ ( k + 1) = Gx ~ ( k) + Hu( k) + K ( k + 1) = (G K C) ( k) { y( k) y~ ( k) } x~ ( k + 1) = (G K C)x ~ ( k) + Hu( k) + K y( k) Obsrvr Fdback Gan Marx 18 9

10 Conrollr Puls Transfr Funcon Marx: U( z) = KX ~ ( z) z-ransform of prdcon obsrvr and subsuon of abov fdback quaon ylds: U(z) = 1 [ K( zi G + K C + HK) K ] Y( z) Conrollr Characrsc quaon: zi G + K C + HK = 0 Ackrman Formula for h Obsrvr Gan Marx: Error Dynamcs Characrsc Equaon 1 C 0 CG 0 K (G) n n 1 = φ, φ(g) = G + α1g + + αn-1g + αn-1i n 1 CG 1 19 Transfr Funcon Approach o Pol placmn-obsrvr Dsgn y( ) = cx( ) ( ) ( x( ) = Ax + bu ) 1 bs ( ) g( s) = c( si A) b = as ( ) ( ) x () = A x + bu() + Ly() c u () = Kx () + r () sx ( s) = A X s + bu ( s) + LY ( s) U( s) = kx ( s) + R( s) ( ) c 20 10

11 U s R s k si A LY s bu s 1 ( ) ( ) = ( c ) [ ( ) + ( )] kadj( si Ac) L kadj( si Ac) b = Y ( s ) U( s) si A si A qs ( ) s ( ) = Y ( s) U ( s) d( s) d( s) c c as ( ) qs ( ) s ( ) as ( ) Y ( s) R( s) = Y ( s) Y ( s) bs ( ) ds ( ) ds ( ) bs ( ) Y ( s) b( s) d( s) b( s) d( s) = = Rs ( ) as ( )[ ds ( ) + ( s)] + q( s) b( s) a( s) p( s) + q( s) bs ( ) 21 Dophann Equaon: as ( ) ps ( ) + qs) ( bs ( ) = cs ( ) Ncssary and Suffcn condon for solvng h Dophann quaons. Closd Loop Sysm ordr and Full ordr obsrvr. Pol Placmn dsgn wh Pol-Zro Cancllaon approach

12 Dophann Equaon: as ( ) = s + a s + a s + + as+ a n n 1 n 2 n 1 n bs ( ) = b s + b s + + bs+ b n 1 cs ( ) = s + c s n 1 n 2 n n 2n 1 2n c s + c p( s) = s + p s + p s + + p s + p n n 1 n 2 n 1 n qs ( ) = q s + q s + + q s + q And, n 1 n 2 n 1 n 2 1 as ( ) ps ( ) + qs) ( bs ( ) = cs ( )