CDS 110b: Lecture 8-1 Robust Stability

Transcription

1 DS 0b: Lct 8- Robst Stabilit Richad M. Ma 3 Fba 006 Goals: Dscib mthods fo psnting nmodld dnamics Div conditions fo obst stabilit Rading: DFT, Sctions Fb 06 R. M. Ma, altch

2 Gam lan: Robst fomanc Tajcto Gnation ontoll ocss Estimato Robst ontol Stabilit: bondd inpts bondd otpts fomanc: kp os small Robstnss: do ths things fo all < Simplifing cas: focs on basic contol loop d Stabilit = intnal stabilit fomanc = wightd snsitivit + - (s) + (s) Robstnss =?? + n 3 Fb 06 R. M. Ma, altch

3 Robst Stabilit W Mltiplicativ Unctaint Modl plant as nominal with additional dnamics givn b W W low nctaint W ( j ) W = fqnc wight = nctaint; qi ( jw) high nctaint allows an dnamics to b instd into th plant an b sd to modl paamt nctaint o nmodld dnamics 3 Fb 06 R. M. Ma, altch 3

4 omplmnta Snsitivit and Robstnss z W w Thm A contoll povids obst stabilit to mltiplicativ ptbations if and onl if wh omplmnta snsitivit fnction W (j) T(j) fo all. T : = = H + Not: this thom gaants stabilit fo an tansf fnction (s) satisfing (j) < allows nmodld dnamics (as wll as paamtic nctaint) 3 Fb 06 R. M. Ma, altch 4 zw Intition: H zw psnts th tansf fnction sn b th wightd nctaint W

5 Modls fo Unctaint Additiv nctaint Mltiplicativ nctaint Fdback nctaint W W W Each modl dscibs a class of pocss dnamics: Additiv: Mltiplicativ: Fdback: Us W to shap th nmodld dnamics; < in all cass Robst stabilit conditions givn b small gain thom ompt tansf fnction aond block and qi that this b < (If not, can choos with to dstabiliz) 3 Fb 06 R. M. Ma, altch 5

6 Exampl: altch Dctd Fan Goal: tack aggssiv tajctois with high accac in psnc of nctaintis fomanc spcs Stabl opation Spd and accac Flap addls Dctd Fan altch dctd fan: Rplicats longitdinal dnamics of an aicaft ontolld b dsae al-tim contol sstm Two TI 40 DSs + two DE Alpha pocssos Actation: fan, wing flap, thst vctoing Snsing: optical ncods on stand Unctaint spcs Nois: wind (wall ffcts), lctical nois Modl nctaint: vaiabl moto dnamics, intial paamts, aodnamic vaiations, tc 3 Fb 06 R. M. Ma, altch 6

7 altch Dctd Fan Modl ( x, ) f f mx && = fcos fsin m && = fsin + fcos mg J&& = f d& mglsin Nonlina Diffntial Eqations () s = + + () s Js ds mgl Tansf Fnction (sid foc to pitch) aamt vals J intia aond pitch axis, kg m m.5 mass of fan, kg 0.5 distanc to flaps, m g 0 gavitational constant, m/sc d 0. damping facto (stimatd) l 0.05 offst of cnt of mass, m Socs of nctaint Unknown foc location, Unknown damping cofficint, d Unmodlld aodnamics Unmodlld stand dflctions Snso nois (x,, ) 3 Fb 06 R. M. Ma, altch 7

8 Stp : Nominal fomanc Rasonabl pfomanc spc kp H small Motivation: small changs in fnc gnat small os an also show that this minimizs th ffct of xtnal distbancs Q: How do w spcif th siz of H A: Us magnitd of Bod plot = sinsoidal spons H = : S + = Snsitivit fnction 3 Fb 06 R. M. Ma, altch 8

9 fomanc Spcification via Wightd Snsitivit H = : + = S Us fqnc wighting to spcif pfomanc ov givn ang Dosn t mak sns to ask fo small o ov all fqncis lant can t tack fnc at xtml high fqncis S Kp S small in this gion ctoff W ( j ) S( j ) W ( j ) W ( j) S( j) 3 Fb 06 R. M. Ma, altch 9

10 Exampl: altch Dctd Fan fomanc (/) ( x, ) f f Js + ds + mgl 0 log(0.05) % tacking o fomanc spcification on H Lss than 5% tacking o p to Hz Lss than 0% tacking o p to 5 Hz W W S( j ) W ( j ) W ( j) S( j) 3 Fb 06 R. M. Ma, altch 0

11 Exampl: altch Dctd Fan fomanc (/) Js + ds + mgl ( s + 5) s () = 0 ( s + 300) W 0 = ( s /+ ) Bod lot 50 W S + W T Q: Dos sstm satisf pfomanc spcification? 0 has (dg); Magnitd (db) W (j) S(j) Fqnc (ad/sc) Fb 06 R. M. Ma, altch

12 Exampl: altch Dctd Fan Robstnss (/) () s () = = s Js ds mgl + ( x, ) f f Robstnss spcification on Allow p to 0% vaiation in location of foc vcto, Foc location can chang dnamicall Rwit nctaint spcification in tms of block: W + = Js + ds + mgl = + Js + ds + mgl Js + ds + mgl = + = ( + ) hoos W = 0., with (s) < W 3 Fb 06 R. M. Ma, altch

13 Exampl: altch Dctd Fan Robstnss (/) z w W s () = Js + ds + mgl Bod lot ( s + 5) s () = 0 ( s + 300) 50 0 W S + W T Q: Sstm is nominall stabl, bt is it obstl stabl? has (dg); Magnitd (db) W T Fqnc (ad/sc) Fb 06 R. M. Ma, altch 3

14 viw: Robst fomanc z W w S( j ) W ( j ) Thm povids obst pfomanc to mltiplicativ nctaint if and onl if max( WS + WT ) < Rmaks Givs conditions fo gaantd obst pfomanc Givn W and W, still nd to find that woks 3 Fb 06 R. M. Ma, altch 4

15 Nqist Intptation of Robst fomanc Stabilit and pfomanc conditions: W - Nots Siz of balls vais as fqnc changs ondition is qivalnt to max( WS + WT ) < WL. Nqist plot otsid nominal stabilit. Nqist plot otsid W ball nominal pfomanc (gnalizs gain/phas magin) 3. Nqist plot otsid W ball obst stabilit 4. ombind non-intsction obst pfomanc 3 Fb 06 R. M. Ma, altch 5

16 Exampl: altch Dctd Fan Dsign oblm (/) ( x, ) f f mx && = m && = J&& = f f f cos sin + f f sin cos mg () s = + + () s Js ds mgl z w W S( j ) W W ( j ) 0 = ( s /+ ) W = 0. 0 W W W 3 Fb 06 R. M. Ma, altch 6

17 Exampl: altch Dctd Fan Dsign oblm (/) 0 W S W T 0 0 W S + W T W S W T W S + W T Sstm satisfis nominal pfomanc Sstm satisfis obst stabilit Sstm dos not hav obst pfomanc Fb 06 R. M. Ma, altch 7

18 Smma z W w S( j ) W ( j ) Thm povids obst pfomanc to mltiplicativ nctaint if and onl if max( WS + WT ) < Rmaks Givs conditions fo gaantd obst pfomanc Givn W and W, still nd to find that woks 3 Fb 06 R. M. Ma, altch 8