Loc al inst a biliti Process Conc lusions yt)-1.6y(t-1)-0.75y(t-2) · Averaging a usefu tod e(t)+1.5e(t-1)+0.75e(t-2) Captures a key feature of adaptive systems The B-pdlynomial has zeros at Parameters change sowly 5o±0.81l1 Simplification · Problemsplit in two C(212)=-0.40±0.4oi Linear system 八~ Nonliear equation of lower dmension f(0,t) Good insight into rob usiness issues good complement to simulation e But still dffialt c K.J. Astrom and B WittenmarkLocal Instabilities Process y(t) ￾ 1:6y(t ￾ 1) ￾ 0:75y(t ￾ 2) = u(t ￾ 1) + u(t ￾ 2) + 0:9u(t ￾ 3) e(t)+1:5e(t ￾ 1) + 0:75e(t ￾ 2) The B-polynomial has zeros at z1;2 = ￾0:50  0:81i and C(z1;2) = ￾0:40  0:40i 0 500 1000 1500 2000 −1 0 1 2 3 Time s^0 r^1 r^2 s^1 Conclusions  Averaging a useful tool  Captures a key feature of adaptive systems { Parameters change slowly  Simplication  Problem split in two { Linear system  = constant { Nonliear equation of lower dimension d dt = f (; t)  Good insight into robustness issues  Good complement to simulation  But still dicult c K. J. Åström and B. Wittenmark 9