FEEDBACK SYSTEMS: ABSOLUTE STABILITY controllable Circle criterion: quadratic common Lyapunov function is strictly positive real (SPR): Common way in which a switched system arises Switched system is GUES Discuss passivity a little more, mention KYP (see notes) More on Popov later For this reduces to SPR (passivity) Popov criterion not suitable: depends on

FEEDBACK SYSTEMS: SMALL-GAIN THEOREM controllable Small-gain theorem: quadratic common Lyapunov function Again, switched system is GUES Can handle norms other than 1 (clear)

TWO-DIMENSIONAL SYSTEMS Necessary and sufficient conditions for GUES known since 1970s worst-case switching Planar geometry: explain briefly worst-case switching – already mentioned earlier quadratic common Lyap fcn <=> convex combinations of Hurwitz