f feed f o n Lyapunov rule MT rule d e (a) d t M Rule () Process ∑)- Lv 叫G Model (c) Is there really mag ic in this world? A First order s Proc ess model a First Order System, co nt dy rivative of lyapunov func tion Desired res po nse dv de M - aMyM+ bMc Cont roller amne+-(b02 +a-am) 62 t yuce t he error Adapt at ion law de aMe-(692+a-am)y+(boo bM) Candidate for Lyapunov fun tion b C K.J. Astrom and BWittenmarkAdaptation of Feedforward Gain Lyapunov rule d dt = ￾ uce MIT Rule d dt = ￾ ye θ Σ – Model Process + Σ – Model Process + y y e e θ uc uc kG(s) kG(s) k0G(s) k0G(s) y m y m − γ s − γ s Π Π Π Π (a) (b) Is there really magic in this world? Simulation MIT rule 0 5 10 15 20 0 1 0 5 10 15 20 0 1 0 5 10 15 20 0 1 Time Time Time (a) ^  (b) ^  (c) ^  Lyapunov rule 0 5 10 15 20 0 1 0 5 10 15 20 0 1 0 5 10 15 20 0 1 Time Time Time (a) ^  (b) ^  (c) ^  A First Order System Process model dy dt = ￾ay + bu Desired response dym dt = ￾amym + bmuc Controller u = 1uc ￾ 2y The error e = y ￾ ym de dt = ￾ame ￾ (b2 + a ￾ am)y + (b1 ￾ bm) uc Candidate for Lyapunov function V (e; 1; 2) = 1 2  e 2 + 1 b (b2 + a ￾ am)2 + 1 b (b1 ￾ bm)2 A First Order System, cont Derivative of Lyapunov function dV dt = e de dt + 1 (b2 + a ￾ am) d2 dt + 1 (b1 ￾ bm) d1 dt = ￾ame 2 + 1 (b2 + a ￾ am)  d2 dt ￾ ye + 1 (b1 ￾ bm)  d1 dt + uce Adaptation law d1 dt = ￾ uce d2 dt = ye c K. J. Åström and B. Wittenmark 5