Part 3 System Identification

Transcription

1 2.6 Sy Idnificaion, Eiaion, and Larning Lcur o o. 5 Apri 2, 26 Par 3 Sy Idnificaion Prpci of Sy Idnificaion Tory u Tru Proc S y Exprin Dign Daa S Z { u, y } Conincy Mod S arg inv θ θ ˆ M θ ~ θ? Ky Quion: Q: I a gin daa inforai noug o uniquy drin a od fro a gin od? Do Z conain ufficin inforaion o diingui any wo od in M? Q2: I ry iniizing V θ good noug o obain ru unbiad od? Wa if ru od i no inod in od? How i od-daa fiing infuncd by noi caracriic and inpu propri? Q3: How accura i iad od? How uc arianc, xpcd rror, c.? How uc daa ndd? How o dign xprin Ky Ru Inforai xprin and prin xciaion Conin unbiad ia Signa o noi raion Aypoic arianc Inpu dign: Pudo Rando Binary igna Accuracy-arianc rad-off Sy ordr ia: Mod cion

2 Maaica oo for Par 3 y idnificaion Dicr Fourir ranfor and pcra anayi Cnra ii or Rando proc: wid-n aionary proc, rgodic proc, c. Frquncy Doain Anayi. Dicr Fourir Tranfor and Powr Spcru Dicr Fourir ranfor of a apd-daa y: x, x + i X x o a X i a 2π -priodic funcion: Ti i + 2 πn i i 2πn + 2 X πn x x X X 2 X + 2 πn X 3 A priodic funcion can b xpandd o a Fourir ri xpanion. Trfor, w can wri π π π 2 π x : i i i i X d x d x d π π π : 4 wic an a inr ranfor of X xi: π in x n X d 2π 5 π Powr Spcru xi: Conidr a driniic, boundd qunc { } for wic foowing ii 2

3 R i + : Coarianc Funcion 6 Arag i T powr pcru of { } i dfind a Fourir ranfor of auo-coarianc funcion R Inr Tranfor: A pcia ca i: i Φ R 7 π i R Φ d 2 8 π π π 2 i i Φ 2π π R d 9 Iporan! W wi ofn u i forua o obain an of a quard igna, a pcia ca of. Wi oi W a n Wi oi in any cion of priou cur. W dfind { } a a qunc of indpndn rando ariab wi zro an au and coarianc : E [ ] ow i caracriic ar foray dfind uing Powr Spcru. T auo-coarianc of rando proc i gin by 3

4 [ ] E i R i + E[ + ] δ T f and id of abo xprion i a i arag, wi rig and id i an nb arag. If wo arag ar a, proc i cad rgodic. W au i rgodiciy for o of proc. S dicuion a nd of i cur no. For abo quaion Powr pcru i gin by i i R Φ δ 2 T figur bow ow po of auo-coarianc R again i and corrponding powr pcru Φ again frquncy. o a powr pcru po i conan for nir frquncy. In opic, i an a ig a a unifor diribuion or nir wa ng, a i, Wi. Ti i wy rando proc i cad Wi oi. R Φ A coord rando igna can b crad wi Wi noi going roug a dynaica proc. T foowing or pay a ajor ro in any of anay inod in y idnificaion. 4

5 Tor L Hq b ranfr funcion of a BIBO ab proc wi a Wi noi inpu of arianc. H powr pcru of i gin by q 3 q G [ ] E Saionary T 2 4 i H Φ wr i i nor of a Proof copx nubr. cobining wo 5 R i i T powr pcru i n gin by R i + orwi : : δ : ax, < for R 5

6 6 Φ ax, R pacing by H R i i i i i i i H H 2 i For a aionary igna wi pcru { } wi pcru Φ w w G q G q w T powr pcru and cro pcru ar gin by i 7 Φ G Φ i 8 Φ G Φ.2 Appying pcra Anayi o Sy Idnificaion w 2 9 y G q u + H q Driniic Socaic Two iu o carify for aaica rigor Sricy paing, proc i no aionary; inpu u dri y. Trfor, coarianc funcion R canno b dfind in gnra. R i T aionary propry a w nd i xinc of i coarianc. Tn, w xnd dfiniion o on auing xinc of R Quai-Saionary wid n aionary Iporan ori and cniqu of y idnificaion ar dpndn upon pcra of inod igna, i.. ony cond-ordr propri, and no on any igr-ordr propri. 6

7 ar ocaic proc. T coarianc funcion for i yp of ariab u b gin an nb an: Driniic and ocaic proc ar ixd. { },{ } 2 R i E[ ] If i i quian o: 2 R i ran wi b ry connin, inc w nd o conidr ony on raizaion of ocaic proc, rar an conidring wo cocion of nb arag. Ti i an rgodiciy prob. For dynaica y, i rgodiciy od for igna gnrad roug unifory ab fir Gq 22 G q Undr i aupion, orica boundary bwn ocaic and driniic proc i ow. 7