1. 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

2. 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

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

4. 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

5. 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:

6. 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