stability under constrained switching
DESCRIPTION
STABILITY under CONSTRAINED SWITCHING. Daniel Liberzon. Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign. TWO BASIC PROBLEMS. Stability for arbitrary switching Stability for constrained switching. - PowerPoint PPT PresentationTRANSCRIPT
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STABILITY under CONSTRAINED SWITCHING
Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
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TWO BASIC PROBLEMS
• Stability for arbitrary switching
• Stability for constrained switching
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MULTIPLE LYAPUNOV FUNCTIONS
Useful for analysis of state-dependent switching
– GAS
– respective Lyapunov functions
is GAS
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MULTIPLE LYAPUNOV FUNCTIONS
decreasing sequence
decreasing sequence
[DeCarlo, Branicky]
GAS
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DWELL TIME
The switching times satisfy
dwell time– GES
– respective Lyapunov functions
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DWELL TIME
– GES
Need:
The switching times satisfy
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DWELL TIME
– GES
Need:
The switching times satisfy
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DWELL TIME
– GES
Need:
must be 1
The switching times satisfy
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AVERAGE DWELL TIME
# of switches on average dwell time
– dwell time: cannot switch twice if
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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
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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
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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:
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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
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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
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STABILIZATION by SWITCHING
– both unstable
Assume: stable for some
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STABILIZATION by SWITCHING
[Wicks et al. ’98]
– both unstable
Assume: stable for some
So for each
either or
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UNSTABLE CONVEX COMBINATIONS
Can also use multiple Lyapunov functions
Linear matrix inequalities