Equilibrium values trans fer function A vera ag ing A nalysis c 61G 62G Control error e(t)=y(t-ynht)=(Gdp)-Gmp))udt) dt Yuo Re f(o, 8, 2)) d62 F(a,61,62) nhiw)G(iw)+gri B.-Imt1/G(iw)l FO Gmi) IGm(iw/G(iw) Disc uss hig h and lo Lo cal stabi accuracy of averaging Linearize the a verged equations Consider t he b= 2, am= bm= 3 G(s A 2 GmI -Gm2cos gm w here arctan(w/am GM(s A4+a(1 Dis c uss convergence rate as a function of CK.J. Astrom and BWittenmarkEquilibrium Values Closed loop transfer function Gc = ^ 1G 1 + ^ 2G Control error e(t) = y(t) ￾ ym(t)=(Gc(p) ￾ Gm(p)) uc(t) Sinusoidal signals ^  0 1G(i!) = ^  0 2Gm(i!)G(i!) + Gm(i!) Hence ^  0 1 = Imf1=G(i!)g Imf1=Gm(i!)g ^  0 2 = ￾ ImfGm(i!)=G(i!)g ImGm(i!) Discuss high and low !, signs etc. Averaging Analysis Ge = ^ 1G 1 + ^ 2G ￾ Gm GT ' =  ￾1 ^ 1G 1 + ^ 2G  d 1 dt = ￾ u2 0 2 Re F (!;  1;  2) d 2 dt = u2 0 2 Re  F (!;  1;  2)  1G(￾i!) 1 +  2G(￾i!) F (!;  1;  2) =  1G(i!) (1 +  2G(i!) ￾ Gm(i!) Accuracy of Averaging Consider the case a = 1, b = 2, am = bm = 3, = 1, uc = sin !t G(s) = 2 s + 1 GM (s) = 2 s + 2 0 20 40 60 80 100 0 1 Time Local Stability Linearize the averaged equations A = u2 0jGmj 2 0 1  ￾ cos m jGmj cos 2m jGmj ￾jGmj2 cos m  where m = arctan(!=am) 2 + (1 + cos2 m) +  2 sin2 m = 0 where  = u2 0amb 2 (a2m + !2) Discuss convergence rate as a function of frequency. c K. J. Åström and B. Wittenmark 5