An Example G(s +a)(s+ Stability co ndit io ns ko b2 Analysis of MRAs for First Order MIT- b2 System AspR k b(ab-w2) Equilibrium conditions Local st ability (a) ce rate ● Ro bust ness The system Design Mod Desired Respo nse Gm(s)=bm Anal - yuce Cont roller u(t)=6 dt rye Lyapunov design 叫G(s) y= G(pju Gm(puc u=61ue-62y C K.J. Astrom and BWittenmarkAn Example Gm(s) = a s + a G(s) = ab (s + a)(s + b) Stability conditions  MIT = k0 k b 2 + !2 b2 SPR = k0 k a(b 2 + !2 ) b(ab ￾ !2 ) ! < p ab 0 100 300 500 0.0 0.5 1.0 0 100 300 500 0.0 0.5 1.0 0 100 300 500 0 5 10 0 100 300 500 0 5 10 Time Time Time Time (a) (b) (c) (d) ^  ^  ^  ^  Analysis of MRAS for First Order System  Equilibrium conditions  Local stability  Convergence rate  Robustness The System Design Model G(s) = b s + a Desired Response Gm(s) = bm s + am Controller u(t) = 1uc ￾ 2y Lyapunov design − Σ Π + e u y Σ Π Π Π − + uc Gm (s) G(s) θ 1 θ 2 γ s − γ s Analysis d^ 1 dt = ￾ uce d^ 2 dt = ye e = y ￾ ym y = G(p)u ym = Gm(p)uc u = ^ 1uc ￾ ^ 2y c K. J. Åström and B. Wittenmark 4