SWITCHED SYSTEMS Switched system: is a family of systems is a switching signal Assuming common equilibrium Sigma is piecewise constant Hybrid systems give rise to classes of switching signals Measurable switching – related to control system (Filippov’s lemma), more on this later : stability Properties of the continuous state

STABILITY ISSUE Asymptotic stability of each subsystem is unstable And vice versa: can stabilize, even if subsystems are unstable This is valid in dimensions 2 and higher Asymptotic stability of each subsystem is not sufficient for stability

TWO BASIC PROBLEMS Stability for arbitrary switching Stability for constrained switching Second is more relevant for control First leads to second

TWO BASIC PROBLEMS Stability for arbitrary switching Stability for constrained switching In this talk we’ll mainly deal with the first problem, except for towards the end

GLOBAL UNIFORM ASYMPTOTIC STABILITY GUAS is Lyapunov stability plus asymptotic convergence Reduces to standard GAS for non-switched systems Explain uniformity w.r.t. sigma GUES:

GUAS and COMMON LYAPUNOV FUNCTIONS GUES: is GUAS if (and only if) s.t. Effectively two ways: prove GUAS directly or find V Uniformity – with respect to sigma! some technical conditions are needed for “only if” need W to guarantee uniformity where is positive definite quadratic is GUES