STABILITY under CONSTRAINED SWITCHING Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign
STABILITY under CONSTRAINED SWITCHING Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign.
Dec 18, 2015
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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
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