a Minor extensi where the transfer function G1 is SPR The errore=y- ym can then be witten as MRAS with aug mented error e=G(8-0 )uc=(G1G2( 0-8 )uc Model G1(G2(0-6)uc+(0-")G2uc-(0-6)G2uo Introduce Process -(@-引° where n is the error augmentation defi ned by G1(6-6°)G Use adaptation law dt Compare Str and Mras W hat You should Know MRAS d e e the ideas Ypf How to make abstractions pf=-Gf(p)grade(t) The( E=GSPR(y-ym)+n=G SpRe +n Nations df g ain phase and passivity Direct str PR and sPr y(t)=pf(t-do)8 The key resuts The small gain theorem e(t)=v(t)-=y(t)-9(t-do)(t-1) The passivity theorem (t)=6(t-1)+P(t)9f(t-do)e(t) The arde criterion Abilit impute g e(t=yt-y(t) Determne passway =y(t)-ym(t)+ym()-y(t Apply to adaptive cortrd e(t)+、(t) Similarities between MRAS STIRA Minor Extension Factor G = G1G2 where the transfer function G1 is SPR. The error e = y ￾ ym can then be written as e = G( ￾  0 )uc = (G1G2)( ￾  0 )uc = G1￾G2( ￾  0 )uc + ( ￾  0 )G2uc ￾ ( ￾  0 )G2uc
Introduce " = e +  where  is the error augmentation dened by  = G1( ￾  0 )G2uc ￾ G( ￾  0 )uc = G1(G2uc) ￾ Guc Use adaptation law d dt = ￾ G2uc MRAS with Augmented Error – + Model Process y – e Σ Σ θ Σ θ + η ε uc ym k0G kG − γ s G2 G1 u c Π Π Π Compare STR and MRAS MRAS d dt = 'f " 'T f = ￾Gf (p)grad"(t) " = GSPR(y ￾ ym) +  = GSPRe +  Direct STR y(t) = ' T f (t ￾ d0)  "(t) = y(t) ￾ =^ y(t) ￾ 'T f (t ￾ d0) ^ (t ￾ 1) ^ (t) = ^ (t ￾ 1) + P (t)' T f (t ￾ d0) "(t) Residual "(t) = y(t) ￾ y^(t) = y(t) ￾ ym(t) + ym(t) ￾ y^(t) = e(t) + (t) What You Should Know!  The ideas { How to make abstractions  The concepts { Notions of gain phase and passivity { PR and SPR  The key results { The small gain theorem { The passivity theorem { The circle criterion  Abilities { Compute gain { Determine passivity { Apply to adaptive control  Similarities between MRAS STR c K. J. Åström and B. Wittenmark 6