MEM 355 Performance Enhancement of Dynamical Systems A First Control Problem - Cruise Control

Transcription

1 MEM 355 Prformanc Enhancmn of Dynamical Sysms A Firs Conrol Problm - Cruis Conrol Harry G. Kwany Darmn of Mchanical Enginring & Mchanics Drxl Univrsiy

2 Cruis Conrol ( ) mv = F mg sinθ cv v +.2v= u 9.8θ v[ m/ s] sd ( m/s=36 km/h=22 mils/hr) u normalizd hrol u θ [ rad] roadway slo sd command sd rror - conrol driving forc ngin disurbing forc - vhicl sd

3 Criria Wha characrisics do w xc in a good cruis conrol sysm? Ulima racking rror in rsons o sd command Ulima racking rror in rsons o disurbanc Transin rsons im Transin ovrshoo, undrshoo Robusnss olranc o unmodld dynamics or aramr variaion

4 Cruis Conrol: driv rror rsons from diffrnial quaions Assumions: 'roorional' lus 'ingral' conrol, ignor ngin dynamics ( ) i ( ( ) ( )) ( ) ( ) ( ) u = k v v + k v τ v τ dτ ( ) = ( ) ( ) Dfin: v v v +.2v= u 9.8θ +.2 = u θ + v +.2v ( ) ( ( τ) ) ( ) ( ) u = k v v + k v v dτ u = k + k i i + (.2 + k) + k i = 9.8 θ + v +.2v ( s + ).2 9.8s E( s) = V ( s) + Θ s s k s+ k s k s+ k ( ) ( ) 2 2 i i Error rsons o command ( ) sd rror 2 Inus: v θ sd command road disurbanc Error rsons o disurbanc

5 Cruis Conrol: driv rror rsons using block diagrams Engin Comnsaor θ 9.8 ( ) Vhicl v Rmmbr h formula Or do h algbra ks+ k s i s +.2 ( s +.2) v : Gv = G = ks+ k s + ( + k ) s+ k + s s+.2 ks + ki θ : E = ( 9.8Θ) E s+.2 s ks + ki s + E = Θ Gθ = 2 s s+.2 s k s+ ki v 2 i.2 i ( ) v Closd loo ransfr funcions wih wo dsign aramrs

6 Cruis Conrol Rsons o uni s disurbanc 9.8s E( s) = Θ( s), Θ ( s) = s (.2 ) 2 s + + k s+ ki ( s + ).2 E( s) = V ( s), V ( s) = s (.2 ) 2 s + + k s+ ki Rsons o uni s command k =, k =,.5,, 2, 6 i

7 Obsrvaions Boh ransfr funcions hav sam dnominaor (sam ols), bu diffrn numraors (diffrn zros) Whn = (roorional conrol) h ulima rror is no zro, k i in fac h ulima rror in disurbanc is larg. rsons o command is vry small, bu o For sabiliy w can look a ihr ransfr funcion, bu for rformanc w nd o considr boh. To valua k, k i is hlful o mak h associaion i s + (.2 + k ) s+ k s + 2ρω s+ ω i

8 Rfin h Conrol Noic ha w can scify, ρω, and choos k i = ω, k = 2ρω.2 2 L us fix ρ =.77 and look a ω =, 2, 2, Disurbanc rsons Command rsons

9 Effc of Engin Dynamics G ng Error rsons o command s using h fass conrollr. ρ =.77, ω = 3 Engin im consan. sc, and.2 sc. 2 =, τ =.,.2 τ s τ =.sc τ =.2 sc Dramaic rducion in daming raion Unsabl

10 Effc of Engin Dynamics ~ 2 Suos w us h slows conrollr ρ =.77, ω = Hr w s how h rsons dgrads whn slow ngin is includd, a las i is sill sabl Pushing for high rformanc ofn lads o non-robus dsign..6 Wih slow ngin Wih idal ngin Error rsons o disurbanc

11 Add Fdforward v Gc ks+ k s i G θ ( ) 9.8 Us il snsor (acclromrs) o sima roadway slo. Gv s +.2 v.6.4 E ( G ) Gv = + GGG v c D.2 Slow conrollr, slow ngin Wih fdforward

12 Summary Any on of h closd loo ransfr funcions can b usd for sabiliy analysis (all hav sam ols) Prformanc analysis usually rquirs considring wo or mor closd loo ransfr funcions. Ulima rror dnds on conrollr y,.g. PI conrollr rsuld in zro rror vnually, bu P conrollr lf som rsidual rror nonrivial in h cas of disurbanc. W can choos conrol aramrs o sha ransin rsons (loca closd loo ols) in his scial cas w usd our knowldg of 2 nd ordr sysm bhavior. Th sysm may b snsiiv o modl accuracy, including nglcd dynamics vn o h oin of insabiliy.

13 Nx Ss Conrolling h ulima rror Shaing h ransin rsons: closd loo ol locaion via roo locus Evaluaing closd loo sysm sabiliy robusnss h abiliy o rmain sabl whn h modl is uncrain Conrollr dsign o achiv robus rformanc Ohr (mor dirc) mhods for shaing ransin rsons