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1
STABILITY OF SWITCHED SYSTEMS
Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
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SWITCHED vs. HYBRID SYSTEMS
: stability
Switching:• state-dependent or time-dependent• autonomous or controlled
Properties of the continuous state
Switched system:
• is a family of systems
• is a switching signal
Details of discrete behavior are “abstracted away”
Hybrid systems give rise to classes of switching signals
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STABILITY ISSUE
unstable
Asymptotic stability of each subsystem is
not sufficient for stability
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TWO BASIC PROBLEMS
• Stability for arbitrary switching
• Stability for constrained switching
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TWO BASIC PROBLEMS
• Stability for arbitrary switching
• Stability for constrained switching
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GUAS and COMMON LYAPUNOV FUNCTIONS
where is positive definite
quadratic is GUES
GUAS:
GUES:
is GUAS if (and only if) s.t.
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COMMUTING STABLE MATRICES => GUES
quadratic common Lyap fcn [Narendra & Balakrishnan ’94]:
0)0()1...(1)1...(2
x
ssAe
ttAe kk
IAPPAT 1111
12222 PAPPAT
)(tx )0(xktAe 2 ksAe 1 11sAe12tAe…
1221},2,1{ AAAAP
t1 12 2
1s 2s1t 2t…
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COMMUTATION RELATIONS and STABILITY
GUES and quadratic common Lyap fcn guaranteed for:
• nilpotent Lie algebras (suff. high-order Lie brackets are 0)e.g.
• solvable Lie algebras (triangular up to coord. transf.)
• solvable + compact (purely imaginary eigenvalues)
Lie algebra:
Lie bracket:
Further extension based only on Lie algebra is not possible[Agrachev & L ’01]
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SWITCHED NONLINEAR SYSTEMS
• Global results beyond commuting case – ???
• Commuting systems
• Linearization (Lyapunov’s indirect method)
=> GUAS [Mancilla-Aguilar,Shim et al., Vu & L]
[Unsolved Problems in Math. Systems and Control Theory]
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SPECIAL CASE
globally asymptotically stable
Want to show: is GUAS
Will show: differential inclusion
is GAS
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OPTIMAL CONTROL APPROACH
Associated control system:
where
(original switched system )
Worst-case control law [Pyatnitskiy, Rapoport, Boscain, Margaliot]:
fix and small enough
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MAXIMUM PRINCIPLE
is linear in
at most 1 switch
(unless )
GAS
Optimal control:(along optimal trajectory)
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SYSTEMS with SPECIAL STRUCTURE
• Triangular systems
• Feedback systems
• passivity conditions
• small-gain conditions
• 2-D systems
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TRIANGULAR SYSTEMS
exponentially fast
0
exp fast
quadratic common Lyap fcn diagonal
Need to know (ISS)
For nonlinear systems, not true in general
For linear systems, triangular form GUES
[Angeli & L ’00]
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FEEDBACK SYSTEMS: ABSOLUTE STABILITY
Circle criterion: quadratic common Lyapunov function
is strictly positive real (SPR):
For this reduces to SPR (passivity)
Popov criterion not suitable: depends on
controllable
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FEEDBACK SYSTEMS: SMALL-GAIN THEOREM
Small-gain theorem:
quadratic common Lyapunov function
controllable
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TWO-DIMENSIONAL SYSTEMS
quadratic common Lyap fcn <=>
convex combinations of Hurwitz
Necessary and sufficient conditions for GUES
known since 1970s
worst-caseswitching
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WEAK LYAPUNOV FUNCTION
Barbashin-Krasovskii-LaSalle theorem:
• (weak Lyapunov function)
• is not identically zero along any nonzero solution
(observability with respect to )
observable=> GAS
Example:
is GAS if s.t.
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COMMON WEAK LYAPUNOV FUNCTION
To extend this to nonlinear switched systems and
nonquadratic common weak Lyapunov functions,
we need a suitable nonlinear observability notion
Theorem: is GAS if
• .
• observable for each
• s.t. there are infinitely many
switching intervals of length
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NONLINEAR VERSION
Theorem: is GAS if s.t.
• s.t. there are infinitely many
switching intervals of length
•
• Each system
is small-time norm-observable:
pos. def. incr. :
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TWO BASIC PROBLEMS
• Stability for arbitrary switching
• Stability for constrained switching
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MULTIPLE LYAPUNOV FUNCTIONS
)(,)( 21 xfxxfx GAS
21 , VV respective Lyapunov functions
t1 12 2
)()( tV t
Useful for analysis of state-dependent switching
)(xfx is GAS
=>
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MULTIPLE LYAPUNOV FUNCTIONS
t1 12 2
)()( tV t
decreasing sequence
decreasing sequence
[DeCarlo, Branicky]
=> GAS
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DWELL TIME
The switching times satisfy...,, 21 tt Dii tt 1
dwell time )(,)( 21 xfxxfx GES
21 , VV respective Lyapunov functions
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DWELL TIME
The switching times satisfy...,, 21 tt Dii tt 1
)(,)( 21 xfxxfx GES
,||)(|| 211
21 xbxVxa )()( 111
1 xVxfxV
,||)(|| 222
22 xbxVxa )()( 222
2 xVxfxV
t1 12
1t 2t0t
Need: )()( 0121 tVtV
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DWELL TIME
The switching times satisfy...,, 21 tt Dii tt 1
)(,)( 21 xfxxfx GES
,||)(|| 211
21 xbxVxa )()( 111
1 xVxfxV
,||)(|| 222
22 xbxVxa )()( 222
2 xVxfxV
)( 21 tV )( 01)2(
1
2
2
1 1 tVeab
ab D
must be 1Need: )()( 0121 tVtV
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AVERAGE DWELL TIME
AD
tTNtTN
0),(
# of switches on ),( Tt average dwell time
10N dwell time: cannot switch twice if ADtT
00N no switching: cannot switch if ADtT
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AVERAGE DWELL TIME
AD
tTNtTN
0),(
)|(|)()|(| 21 xxVx p
)()( xVxf ppp
xV
qpxVxV qp ,),()(
)(xfx
=> is GAS
if log
AD
Theorem: [Hespanha]
Useful for analysis of hysteresis-based switching logics
GAS is uniform over in this class
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MULTIPLE WEAK LYAPUNOV FUNCTIONS
Theorem: is GAS if
• .
• observable for each
• s.t. there are infinitely many
switching intervals of length
• For every pair of switching times
s.t.
have
– milder than a.d.t.
Extends to nonlinear switched systems as before
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APPLICATION: FEEDBACK SYSTEMS
Theorem: switched system is GAS if
• s.t. infinitely many switching intervals of length
• For every pair of switching times at
which we have
(e.g., switch on levels of equal “potential energy”)
observable
positive real
Weak Lyapunov functions:
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RELATED TOPICS NOT COVERED
• Computational aspects (LMIs, Tempo & L)
• Formal methods (work with Mitra & Lynch)
• Stochastic stability (Chatterjee & L)
• Switched systems with external signals
• Applications to switching control design
33
REFERENCES
Lie-algebras and nonlinear switched systems: [Margaliot & L ’04]
Nonlinear observability, LaSalle: [Hespanha, L, Angeli & Sontag ’03]
(http://decision.csl.uiuc.edu/~liberzon)
