STABILITY under CONSTRAINED SWITCHING

Daniel Liberzon

Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign

TWO BASIC PROBLEMS

• Stability for arbitrary switching

• Stability for constrained switching

MULTIPLE LYAPUNOV FUNCTIONS

Useful for analysis of state-dependent switching

– GAS

– respective Lyapunov functions

is GAS

MULTIPLE LYAPUNOV FUNCTIONS

decreasing sequence

decreasing sequence

[DeCarlo, Branicky]

GAS

DWELL TIME

The switching times satisfy

dwell time– GES

– respective Lyapunov functions

DWELL TIME

– GES

Need:

The switching times satisfy

DWELL TIME

– GES

Need:

The switching times satisfy

DWELL TIME

– GES

Need:

must be 1

The switching times satisfy

AVERAGE DWELL TIME

# of switches on average dwell time

– dwell time: cannot switch twice if

AVERAGE DWELL TIME

Theorem: [Hespanha ‘99] Switched system is GAS if

Lyapunov functions s.t. • .

•

•

Useful for analysis of hysteresis-based switching logics

# of switches on average dwell time

MULTIPLE WEAK LYAPUNOV FUNCTIONS

Theorem: is GAS if

• .

•

•

•

– milder than ADT

Extends to nonlinear switched systems as before

observable for each

s.t. there are infinitely many

switching intervals of length

For every pair of switching times

s.t.

have

APPLICATION: FEEDBACK SYSTEMS (Popov criterion)

Corollary: switched system is GAS if

• s.t. infinitely many switching intervals of length

• For every pair of switching times at

which we have

linear system observable

positive real

See also invariance principles for switched systems in: [Lygeros et al., Bacciotti–Mazzi, Mancilla-Aguilar, Goebel–Sanfelice–Teel]

Weak Lyapunov functions:

STATE-DEPENDENT SWITCHING

But switched system is stable for (many) other

Switched system

unstable for some

no common

switch on the axes

is a Lyapunov function

STATE-DEPENDENT SWITCHING

But switched system is stable for (many) other

level sets of level sets of

Switched system

unstable for some

no common

Switch on y-axis

GAS

STABILIZATION by SWITCHING

– both unstable

Assume: stable for some

STABILIZATION by SWITCHING

[Wicks et al. ’98]

– both unstable

Assume: stable for some

So for each

either or

UNSTABLE CONVEX COMBINATIONS

Can also use multiple Lyapunov functions

Linear matrix inequalities