Com parison with mit rule A First Order System, cont abnor yapunav function V(e,61,6 2+(2+2-a2+(1-h)2 Its derivative MIT rle Negative semi-definite but nct negative definite de V=-2am edf 2ame-ame(b2+a-an by+(b1-bn) u) Error goes to zero but parameters do nat necessanly go to tHar correct values! simulation Process inputs and outputs State feed back ame idea Parameters Denve error eq uation 01 Find Lyapunov equation c K.J. Astrom and B Wit tenmaA First Order System, cont The Lyapunov function V (e; 1; 2) = 1 2  e 2 + 1 b (b2 + a ￾ am)2 + 1 b (b1 ￾ bm)2 Its derivative dV dt = ￾ame 2 Negative semi-denite but not negative denite V = ￾2ame de dt = ￾ 2ame (￾ame ￾ (b2 + a ￾ am)y + (b1 ￾ bm) uc) Error goes to zero but parameters do not necessarily go to their correct values! Comparison with MIT rule Lyapunov − Σ Π + e u y Σ Π Π Π − + uc Gm (s) G(s) θ 1 θ 2 γ s − γ s MIT rule − Σ Π + e u y Σ Π Π Π − + uc Gm (s) G(s) θ 1 θ 2 γ s − γ s am s + am am s + am Simulation Process inputs and outputs 0 20 40 60 80 100 −1 1 0 20 40 60 80 100 −5 0 5 Time Time (a) ym y (b) u Parameters 0 20 40 60 80 100 0 2 4 0 20 40 60 80 100 −1 1 Time Time 1 2 = 5 = 1 = 0:2 = 5 = 1 = 0:2 State Feedback Same idea  Find a controller structure  Derive error equation  Find Lyapunov equation c K. J. Åström and B. Wittenmark 6