MM1. Introduction to State-Space Method

Transcription

1 MM Itroductio to Stt-Spc Mthod Wht tt-pc thod? How to gt th tt-pc dcriptio? 3 Proprty Alyi Bd o SS Modl Rdig Mtril: FC: p C: p- /4/8 Modr Cotrol Wht th SttS tt-spc Mthod? I th tt-pc thod th dyic yt i dcrid y t of firt-ordr diffrtil/diffrc qutio i th vctor-vlud tt d th olutio i viulizd trjctory of thi tt vctor i pc Advtg: To tudy or grl odl To dl with ultil iput d ultipl output yt To coct itrl d xtrl dcriptio /4/8 Modr Cotrol

2 Stt-Spc Spc criptio Th tt-pc rprttio i giv y th qutio: d t A t Bu t dt Y t C t u t Stt qutio Output qutio whr t i x vctor rprtig th tt g poitio d vlocity vril i chicl yt ut i clr rprtig th iput yt i clr rprtig th output Th tric A x B x d C x dtri th rltiohip tw th tt d iput d output vril Stt pc rprttio c lo ud for yt with ultipl iput d output MIMO /4/8 Modr Cotrol 3 How to gt Stt Spc odl? Vi odllig tchiqu Vi trfr fuctio 3 Vi othr SS dcriptio 4 Vi yt idtifictio /4/8 Modr Cotrol 4

3 How to gt Stt Spc odl? Vi odllig tchiqu iffrtil qutio Vi trfr fuctio - rliztio Cotrol coicl: [ABC] tfnumen 3 Vi othr SS dcriptio yt yt 4 Vi yt idtifictio idt i Mtl /4/8 Modr Cotrol 5 Modlig C Motor Workig chi of C otor T t θ t torqu cott i rtur currt lctrootiv forc f cott iffrtil qutio dcriptio J θ θ di R i v dt iplifid : t J θ R t i θ i θ R t v /4/8 Modr Cotrol 6

4 /4/8 Modr Cotrol 7 How to gt Stt Spc odl? Vi odllig tchiqu Vi trfr fuctio 3 Vi othr SS dcriptio 4 Vi yt idtifictio /4/8 Modr Cotrol 8 Cotrol coicl for Stt Stt-Spc criptio Spc criptio [ ] whr C B A U C Y BU A M M O M M

5 /4/8 Modr Cotrol 9 Expl: Modl coicl for G Modl Coicl Rliztio S5 Modl Coicl Rliztio S5 [ ] 3 4 C B A U C Y U B A Syt pol ppr th lt log th digol of A Th urtor tr i th prtil-frctio xpio ppr i th C /4/8 Modr Cotrol Cotrol coicl: [ABC] tfnumen Modl coicl for: CSYS cosystype cy coy'odl' rtur rliztio cy i odl for tht i whr th rl igvlu ppr o th digol of th trix d th coplx cojugt igvlu ppr i -y- lock o th digol of A TYPE: 'copio': G 3 Mtl Iplttio 3 Mtl Iplttio Nu[ ]; [ 7 ]; [ABC]tfNu SyABC CycoSy 'odl' CycoSy 'copio'

6 How to gt Stt Spc odl? Vi odllig tchiqu Vi trfr fuctio 3 Vi othr SS dcriptio 4 Vi yt idtifictio /4/8 Modr Cotrol 3 Trfortio tw SS Modl Pricipl: Stt coordit trfortio for tt-pc odl yt yt /4/8 Modr Cotrol

7 3 Trfor to Cotrol Coicl For Stp o: coputr th cotrollility trix T C [A AB A B A - B ] ctrab ctry Stp two: coputr th row t through t [ ]T - C Stp thr: coputr th tir trfor trix T ta ta M t Stp four: coputr w yt tric uig T d T - A c T AT B c T B C c CT c /4/8 Modr Cotrol 3 3 Trfor to Modl Coicl For Eigvlu d igvctor of trix A λ i ν i A ν i for i Trfor to odl coicl for A BU Y C U A Y C B U U Stp : clcult th igvlu d igvctor of A Stp : cotruct th trfor trix with prtr α i T[α ν α ν α ν ] Stp 3: lct th cl fctor α i of th igvctor uch tht ll th lt of B r uity: B T - B Stp 4: u th dtrid T to clcult A T AT B T B C CT /4/8 Modr Cotrol 4

8 How to gt Stt Spc odl? Vi odllig tchiqu Vi trfr fuctio 3 Vi othr SS dcriptio 4 Vi yt idtifictio /4/8 Modr Cotrol 5 4 Syt Idtifictio for SS odl Wht i Syt Idtifictio? Syt Idtifictio llow you to uild thticl odl of dyic yt d o urd dt How i tht do? Etilly y djutig prtr withi giv odl util it output coicid wll poil with th urd output Wht th CA tool? idt i Mtl Syt idtifictio cour i IRS7 tr /4/8 Modr Cotrol 6

9 Stt-Spc Spc criptio icrt C Th ZOH tchiqu c ployd to gt th tt pc dcriptio of dicrt ti yt A BU k Y C U Y k C whr T : plig priod Φ k k U Γ U k k Φ AT Γ T A τ d τ B Mtl iplttio: SYS cdabctmetho /4/8 Modr Cotrol 7 Aly Bd o SS Modl Eigvlu of th yt iga Triio zro of th yt tzroabc A BU I A dt Y C U C B Trfr fuctio dcriptio of th yt Stility critrio MIMO yt Siultio SS yt /4/8 Modr Cotrol 8

10 Trfr Fuctio fro Stt Equtio Trfor fuctio Y A BU C U G Y U [ABC] tfnumen [NUMEN] tfabc C I A B Stt-pc dcriptio Y A BU C U Eigvlu of A iga G fqucy rpo dcriptio I A dt C dt I B A Pol of G rootdoitor Triio zro tzroabc Zro of G rooturtor /4/8 Modr Cotrol 9