Local and global behavi Linearize around the equilibrium set dr 11o Analysis of Global Behavior ability boundary 61G(0) 61G(0) 1+62G(0)1+62G(0) G(0)61 +62G(0) This differential equat ion has the solut ion Global behavior 2 Measurement noise mmary 62 Exact Mode l D Step Equilibrium is Stability lo st half line C Step drift alo EQ noise equilib rium set until unst able Equilibrium function equilib rium input dependent Noise makes the syst em drift along the 1 Unstable for equilib rium line C K.J. Astrom and B. WittenmarkLocal and Global Behavior Linearize around the equilibrium set dx dt = u2 0 20 1  ￾1 1 1 ￾1  x Stability boundary θ2 θ1 Global behavior −20 −10 0 10 20 0 10 20 −20 −10 0 10 20 0 10 20 (a) ^ 2 ^ 1 (b) ^ 2 ^ 1 Analysis of Global Behavior d 1 dt = ￾ u2 0 2   1G(0) 1 +  2G(0) ￾ Gm(0) d 2 dt = u2 0 2  1G(0) 1 +  2G(0)   1G(0) 1 +  2G(0) ￾ Gm(0) d 2 d 1 = ￾ G(0) 1 1 +  2G(0) This dierential equation has the solution   2 2 + 2 G(0)  2 +   2 1 = const Measurement Noise 0 0.2 0.4 0.6 0.8 1 −0.2 0.0 0.2 0.4 ^ 2 ^ 1 (a) (b) Noise makes the system drift along the equilibrium line! Summary Input Exact Model Unmodeled Dynamics Step Equilibrium is Stability lost half line for some IC Step & Drift along Drift along EQ noise equilibrium set until unstable Sine Correct Equilibrium function equilibrium input dependent Unstable for high frequencies c K. J. Åström and B. Wittenmark 7