Chper Se Vrile Model The Se Vrile of Dic Se The Se Differeil Eqio Sigl-Flow Grph Se Vrile The Trfer Fcio fro he Se Eqio
Irodcio I he previo chper, we ed plce rfor o oi he rfer fcio odel repreeig lier, ie-ivri, phicl e ilizig loc digr o iercoec e I Chper, we r o lerive ehod of e odelig ig ie-doi ehod I Chper, we will coider phicl e decried h-order ordir differeil eqio Uilizig e of vrile ow e vrile, we c oi e of fir-order differeil eqio The ie-doi e vrile odel led ielf eil o coper olio d li
Tie-Vrig Corol Se Wih he red vilili of digil coper, i i coveie o coider he ie-doi forlio of he eqio repreeig corol e The ie-doi i he heicl doi h icorpore he repoe d decripio of e i er of ie The ie-doi echiqe c e ilized for olier, ievrig, d livrile e e wih everl ip d op igl A ie-vrig corol e i e for which oe or ore of he preer of he e vr fcio of ie For eple, he of iile vrie fcio of ie he fel i epeded drig fligh
Ter Se: The e of dic e i he lle e of vrile clled e vrile o h he owledge of hee vrile, ogeher wih he owledge of he ip for, deerie he ehvior of he e for ie Se Vrile: The e vrile of dic e re he vrile ig p he lle e of vrile h deerie he e of he dic e Se Vecor: If e vrile re eeded o decrie he ehvior of give e, he he e vrile c e coidered he copoe of vecor Sch vecor i clled e vecor Se Spce: The -dieiol pce whoe coordie e coi of he i, i,, i, where,,, re e vrile, i clled e pce Se-Spce Eqio: I e-pce li, we re cocered wih hree pe of vrile h re ivolved i he odelig of dic e: ip vrile, op vrile, d e vrile 4
The Se Vrile of Dic Se The e of e i e of vrile ch h he owledge of hee vrile d he ip fcio will, wih he eqio decriig he dic, provide he fre e d op of he e For dic e, he e of e i decried i er of e of e vrile Ip Sigl Se Op Sigl 5
Se Vrile of Dic Se iiil codiio Ip Dic Se Se Op The e vrile decrie he fre repoe of e, give he pree e, he eciio ip, d he eqio decriig he dic 6
7 The Se Differeil Eqio The e of e i decried he e of fir-order differeil eqio wrie i er of he e vrile,,, eqio B Se differeil A ẋ D Op eqio - op igl C D :direc riio ri C :Op ri; B :ip ri A :Se ri; d d A B d d &
Bloc Digr of he ier, Coio Tie Corol Se D ẋ B d C A ẋ A B C D 8
M Groded, M g Mechicl e decried he fir-order differeil eqio Appied orqe T N - ier veloci v /ec ier poiio v F v M dv M d F M d d d F M 9
Mechicl Eple: M-Sprig Dper A e of e vrile fficie o decrie hi e iclde he poiio d he veloci of he, herefore, we will defie e of M M d d d d d d d d d d ; d d e vrile, M M Wll fricio M K :Sprig co
Eple : Coider he previo echicl e Ae h he e i lier The eerl force i he ip o he e, d he diplcee of he i he op The diplcee i ered fro he eqiliri poiio i he ece of he eerl force Thi e i igle-ip-igle-op e [ ] [ ], C, B, - - A C ; B A The e eqio d he op eqio re i he drd for : Op Eqio Se Eqio - - we hve vecor ri for, I The op eqio i : he ; ; d e defie wo vrile : Thi i ecod order ei e i ivolve wo iegror D D
Elecricl d Mechicl Coerpr Eerg Mechicl Elecricl Kieic M / Ieri 5 v / 5 jω Idcor 5 i Poeil Grvi: gh Sprig: 5 Cpcior 5 Cv Diipive Dper / Fricio 5 Bv eior i
eice, oh Appied volge Crre i v i v i v i v
Idcce, H Appied volge v Crre i i v i di d v d v 4
Cpcice, C F Appied volge Crre i v i d C dv i C d v v i C 5
Elecricl Eple: A C Circi ξ v / i / C d eerg of ; i Cv he ewor c i he ol iiil USE KC he jcio dvc ic C i d di i vc d The op of he e i repreeed : v d d C C d d The op igl i he : v o o i v C i C i C V o - 6
7 Eple : Ue Eqio fro he C circi [ ] [ ] C C - C - - - we hve /,,, Whe The op i
Sigl-Flow Grph Model A igl-flow grph i digr coiig of ode h re coeced everl direced rche d i grphicl repreeio of e of lier relio Sigl-flow grph re ipor for feedc e ece feedc heor i cocered wih he flow d proceig of igl i e V f G θ G G ed Eple : 8 - G G Y Y 8
Mo Gi Forl for Sigl Flow Grph I pplicio, we wih o deerie he reliohip ewee ip d op vrile of he igl flow digr The rice ewee ip ode d op ode i he overll gi ewee P P P ph gi of deeri of forwrd ph grph - of ll idividl loop gi h hee wo ode of gi of ll poile coiio of wo oochig loop - of gi prodc of ll poile coiio of - c - d c f,c d,e,f hee oochig loop cofcor of he h forwrd ph deeri of he grph wih he loop ochig he h forwrd ph reoved, h i, he cofcor i oied fro reovig he loop h och ph P 9
Sigl-Flow Grph Se Model U /C / / / v o -/ -/C C C C C U Vo v C C U V G o o / / / ; / / / ; γ β α
he feedc loop fcor - of he forwrd - ph fcor Soe of N q o o o o q P G P U Y G U Y G U Y G
Phe Vrile For: e iiill coider he forh-order rfer fcio For e vrile,,, 4 ; Ner of iegror eql he order of he e U / / / / Y 4 / / / / o U - 4 - - - Y G Y U 4 4 4
4 4 4 G he eoo ed Eplev of U / / / / o Y 4 - - - - [ ] 4 4 4 4 4 4 4 C - - - - ; ; d d
Alerive Sigl-Flow Grph Se Model Coroller G c 5 Moor d od 6 U I Y G c 5 5 5 6 / 5 / I 6 / U -5 - - - 6 ẋ - - 5 r - 5 [ ] Y 4
5 The Se Vrile Differeil Eqio [ ] - - ; - - - 5 d -,, 5 5 r T Y q T Y / / / - - - -5 for for or Coicl Digol
6 The Se Vrile Differeil Eqio [ ] - - - - - 5 d -,, 5 5 5-5 - 5-6 - r T Y q T Y r
7 The Trfer Fcio fro he Se Eqio Give he rfer fcio G, we oi he e vrile eqio ig he igl-flow grph odel ecll he wo ic eqio [ ] B C B C B X I - A Sice B X A I CX B A X X C B A U Y G U Y U U Y U Φ Φ Φ Φ he igle ip i he igle op d i Te he plce rfor
8 Eercie: E DGD A roo-r drive e for oe joi c e repreeed he differeil eqio, where v veloci, poiio, d i i he corol-oor crre P he eqio i e vrile for d e p he ri for for i v d dv v i i v v d d i v d dv d d v, B, - - A B; A d le, Defie -
9 E: A e c e repreeed he e vecor differeil eqio of eqio 6 of he eoo Fid he chrceriic roo of he e DGD - A ; - De I - A De j j λ λ λ λ λ λ λ λ λ B A ẋ
E7: Coider he prig d how i Figre where M g, N/, d N//ec Fid he e vecor differeil eqio Fid he roo of he chrceriic eqio for hi e DGD De λi - A - λ De - - λ λ ; λ λ - λ λ
E8: The l, low-lide hoverig ove ovig ld dec of ll hip i ver dedig, i priclr, i dvere weher d e codiio The hoverig codiio i repreeed he A ri DGD A - 5 - λ - De λi - A De λ 5 λ λ λ 5 λ ; λ - j ; λ -- - λ j
E9: See he eoo DGD [ ] [ ]z - 5-79 ; -/ z 5 - /, - / / - z
P DGD-EG45: C C v v i id C v v d di i v c c / / -/ - / c The e eqio re : d Selec he e vrile Appl KV c