CDS 101/110: Lecture 7.1 Loop Analysis of Feedback Systems

Transcription

1 CDS 11/11: Lctu 7.1 Loop Analysis of Fdback Systms Novmb Goals: Intoduc concpt of loop analysis Show how to comput closd loop stability fom opn loop poptis Dscib th Nyquist stability cition fo stability of fdback systms Intoduc Nyquist Diagam Fist look at gain magin and phas magin Rading: Åstöm and Muay, Fdback Systms, Chapt 1, Sctions 1.1, 1.2

2 Rviw Fom Last Wk 11 Magnitud Phas (dg) Fquncy (ad/sc) F -c C(s) d u P(s) y -k 2

3 What a th units of a Bod Plot? Bod Plot Units Magnitud: Th odinat (o y-axis ) of magnitud plot is dtmind by 2 log 1 GG iiii Dcibls, nams aft A.G. Bll Phas: Odinat has units of dgs (of phas shift) Th abscissa (o x-axis ) is log 1 (fquncy) (usually, ad/sc) Exampl: simpl fist od systm: GG ss = 1 Singl pol at ss = 1/ττ GG iiii = 1 1iiττωω = 1 1ωω 2 ττ 2 In dcibls: 1ττττ 2 log 1 GG(iiii) = 2 log log 1 1 ωωωω = 1 log 1 ( 1 ωωωω 2 )

4 Bod Plot Units (continud) Exampl (continud): simpl fist od systm: GG ss = 1 Bhavio of magnitud in dcibls: 2 log 1 GG(iiii) ωω 1/ττ 1 log 1 2 ωω = 1/ττ 2(log 1 ωω log 1 ττ) ωω 1/ττ ωω 3dB = 1/ττ is th -3dB half-pow o bak point Pcisly: 1 log 1 2 = 3.13 db Unit DC gain (db) Magnitud dcass at 2 db/dcad fo ωω 1/ττ 1ττττ

5 Bod Plot Units(continud) Exampl (continud): simpl fist od systm: GG ss = 1 Phas (agumnt) of tansf function: GG iiii = 1 1iiiiii = 1 1 iiiiii = actan(ωωωω) Asymptotic Appoximation 1ττττ actan(ωωωω) ωω <.1/ττ ππ 4 (1 log 1 ωω log 1 ττ).1ττ ωω 1/ττ ππ ωω > 1/ττ 2

6 Bod Plot (continud)

7 Loop Analysis d C(s) Fist consid simpl unity fdback Pfomanc? Tac how sinusoidal signals popagat aound th closd loop systm. u P(s) y unity gain contoll Altnativ: look fo conditions on PC that lad to instability E.g. : if PC(s) = fo som s = iω, thn systm is not asymptotically stabl Condition on PC is usful bcaus w can dsign PC(s) by choic of C(s) Howv, chcking PC(s) = is not nough; nd mo sophisticatd chck Dos signal gow o dcay? Can b dtmind fom fquncy spons. How do opn loop dynamics ffct closd loop dynamics? HH yyyy = PPPP 1PPPP = nn ppnn cc dd pp dd cc nn pp nn cc Phas (dg) Magnitud (db) P(s)C(s) Pols of H y = zos of 1 PC Fquncy (ad/sc) 7

8 Gam Plan: Fquncy Domain Dsign Goal: figu out how to dsign C(s) so that 1C(s)P(s) is stabl and w gt good pfomanc Pols of H y = zos of 1 PC Would also lik to shap H y to spcify pfomanc at diffnct fquncis Magnitud (db) PPPP 1 Bod Diagam Low fquncy ang: PPPP PPPP 1 1 PPPP 1 (good tacking) 5-45 Bandwidth: fquncy at which closd loop gain = 1 2 opn loop gain 1 Phas (dg) Ida: us C(s) to shap PC (und ctain constaints) Fquncy (ad/sc) Nd tools to analyz stability and pfomanc fo closd loop givn PC 8

9 Nyquist Cition: Wam up Lt th loop tansf function b LL(ss) = PP(ss)CC(ss) Injct sinusoid of fquncy ωω at pt. A. Signal at pt. B has fquncy ωω Oscillatoy signal is slf-maintaining if signal at B is sam as signal at A. This can occu if th is a fquncy ωω such LL iiωω = 1. citical point : whn loop tansf function = Naïv stability ida: LL iω 1 Amplitud of signal at B is lss than amplitud of injctd signal at A. Rality is a bit mo complicatd 9

10 d Nyquist Plot C(s) P(s) A diffnt psntation of fquncy spons of opn loop tansf function, L(s) = P(s)C(s). Fomd by tacing ss aound th Nyquist D contou, Γ u y Nyquist D contou Tak limit as, R Tac fom 1 to 1 along imaginay axis Nyquist Contou (Γ): Imaginay axis Smi-Cicl, o ac, at infinity that conncts ndpoints of imaginay axis Th imag of LL(ss) as ss tavss Γ is th Nyquist plot Not, potion of plot cosponding to ωω < is mio imag of ωω > Nyquist Contou (Γ): If pol of LL(ss) on jjjj-axis, thn cat small smi-cicula dtou aound th pol in RHP. Tak limit as smi-cicl adius Goal: fom complx analysis, w tying to find numb of xcss zos in RHP, which lads to instability 1

11 Nyquist Cition C(s) d Dtmin stability fom (opn) loop tansf function, L(s) = P(s)C(s). Us pincipl of th agumnt fom complx vaiabl thoy (s ading) u P(s) y Imag j -j R Ral Nyquist D contou Tak limit as, R Tac fom 1 to 1 along imaginay axis Thm (Nyquist). Consid th Nyquist plot fo loop tansf function L(s). Lt P # RHP pols of L(s) N # clockwis nciclmnts of Z # RHP zos of 1 L(s) Thn Z = N P To: Y(1) Nyquist Diagams Fom: U(1) iωω < ωω=i ωω=-i iωω > Ral Axis = N=2 Tac fquncy spons fo L(s) along th Nyquist D contou Count nt # of clockwis nciclmnts of th point 11

12 Simpl Intptation of Nyquist C(s) d u P(s) y Basic ida: avoid positiv fdback If L(s) has 18 phas (o gat) and gain gat than 1, thn signals a amplifid aound loop Us whn phas is monotonic Gnal cas quis Nyquist Can gnat Nyquist plot fom Bod plot flction aound al axis 1 Bod Diagams Fom: U(1) 3 Nyquist Diagams Fom: U(1) 2 Phas (dg); Magnitud (db) To: Y(1) Imaginay Axis To: Y(1) 1-2 ωω=- ωω= ωω= Fquncy (ad/sc) ambod(sys) [o bod(sys) in db] Ral Axis amnyquist(sys) 12

13 Exampl: Popotional Intgal* spd contoll d C(s) u P(s) y Rmaks N =, P = Z = (stabl) Nd to zoom in to mak su th a no nt nciclmnts Not that w don t hav to comput closd loop spons 13

14 Mo complicatd systms What happns whn opn loop plant has RHP pols? 1 PC has singulaitis insid D contou ths must b takn into account 1 Pol-zo map 1.5 Nyquist Diagams Fom: U(1) Imag Axis Imaginay Axis To: Y(1) Ral Axis Ral Axis unstabl pol N =, P = 1 Z = NP = (stabl) 14

15 Commnts and cautions Why is th Nyquist plot usful? Old answ: asy way to comput stability (bfo computs and MATLAB) Ral answ: givs insight into stability and obustnss; vy usful fo asoning about stability Nyquist plots fo systms with pols on th jω axis ωω= - chos contou to avoid pols on axis nd to cafully comput Nyquist plot at ths points valuat H(εi) to dtmin diction ωω =i ωω=-i ωω= Cautions with using MATLAB MATLAB dosn t gnat potion of plot fo pols on imaginay axis Ths must b dawn in by hand (mak su to gt th ointation ight!) 15

16 Robust stability: gain and phas magins Nyquist plot tlls us if closd loop is stabl, but not how stabl 3 2 Nyquist Diagam Gain magin How much w can modify th loop gain and still hav th systm b stabl Dtmind by th location wh th loop tansf function cosss 18 phas 1-2 Phas magin How much phas dlay can b adddd whil systm mains stabl Dtmind by th phas at which th loop tansf function has unity gain Bod plot intptation Look fo gain = 1, 18 phas cossings MATLAB: magin(sys) Phas (dg); Magnitud (db) Fquncy (ad/sc) Bod Diagam Gm=7.5 db (at ad/sc), Pm= dg. (at ad/sc)

17 Exampl: cuis contol d C(s) u P(s) y -G(s) Effct of additional snso dynamics Nw spdomt has pol at s = 1 (vy fast); poblms dvlop in th fild What s th poblm? A: insufficint phas magin in oiginal dsign (not obust) 1 Bod Diagam 15 1 Nyquist Diagam Nyquist plots 5 5 Magnitud (db) -5 Imaginay Axis Nyquist Diagam Ral Axis Imaginay Axis Phas (dg) Fquncy (ad/sc) Ral Axis 17

18 Pviw: contol dsign d C(s) u P(s) y -G(s) Appoach: Incas phas magin Incas phas magin by ducing gain can accommodat nw snso dynamics Tadoff: low gain at low fquncis lss bandwidth, lag stady stat o 1 5 Bod Diagam 15 1 Nyquist Diagam Nyquist plots 5 Magnitud (db) -5 Imaginay Axis Nyquist Diagam Ral Axis Imaginay Axis Phas (dg) Fquncy (ad/sc) Ral Axis 18

19 Summay: Loop Analysis of Fdback Systms C(s) Nyquist citia fo loop stability Gain, phas magin fo obustnss d u P(s) y Phas (dg); Magnitud (db) Bod Diagam Gm=7.5 db (at ad/sc), Pm= dg. (at ad/sc) j R Thm (Nyquist). P # RHP pols of L(s) N # CW nciclmnts Z # RHP zos Nyquist Diagam Z = N P -j