TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta,...
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Transcript of TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta,...
TOWARDS ROBUST LIE-ALGEBRAIC STABILITY
CONDITIONS for SWITCHED LINEAR SYSTEMS
49th CDC, Atlanta, GA, Dec 2010
Daniel LiberzonUniv. of Illinois, Urbana-Champaign, USA
Yuliy BaryshnikovBell Labs, Murray Hill, NJ, USA
Andrei A. AgrachevS.I.S.S.A., Trieste, Italy
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SWITCHED SYSTEMSSwitched system:
• is a family of systems
• is a switching signal
Switching can be:
• State-dependent or time-dependent• Autonomous or controlled
Details of discrete behavior are “abstracted away”
: stabilityProperties of the continuous state
Discrete dynamics classes of switching signals
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STABILITY ISSUE
unstable
Asymptotic stability of each subsystem is
not sufficient for stability under arbitrary switching
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KNOWN RESULTS: LINEAR SYSTEMS
Lie algebra w.r.t.
No further extension based on Lie algebra only
Quadratic common Lyapunov function exists in all these cases
Stability is preserved under arbitrary switching for:
• commuting subsystems:
[Narendra–Balakrishnan,
• nilpotent Lie algebras (suff. high-order Lie brackets are 0)e.g.
Gurvits,
• solvable Lie algebras (triangular up to coord. transf.)
Kutepov, L–Hespanha–Morse,
• solvable + compact (purely imaginary eigenvalues)
Agrachev–L]4 of 11
KNOWN RESULTS: NONLINEAR SYSTEMS
• Linearization (Lyapunov’s indirect method)
Can prove by trajectory analysis [Mancilla-Aguilar ’00]
or common Lyapunov function [Shim et al. ’98, Vu–L ’05]
• Global results beyond commuting case
Recently obtained using optimal control approach
[Margaliot–L ’06, Sharon–Margaliot ’07]
• Commuting systems
GUAS
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REMARKS on LIE-ALGEBRAIC CRITERIA
• Checkable conditions
• In terms of the original data
• Independent of representation
• Not robust to small perturbations
In any neighborhood of any pair of matricesthere exists a pair of matrices generating the entire Lie algebra [Agrachev–L ’01]
How to capture closeness to a “nice” Lie algebra?
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DISCRETE TIME: BOUNDS on COMMUTATORS
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– Schur stable
Let
GUES
Idea of proof: take , , consider
GUES:
DISCRETE TIME: BOUNDS on COMMUTATORS
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– Schur stable
GUES:
Let
GUES
Idea of proof: take , , consider
DISCRETE TIME: BOUNDS on COMMUTATORS
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inductionbasis
contraction small
– Schur stable
GUES:
Let
GUES
Idea of proof: take , , consider
DISCRETE TIME: BOUNDS on COMMUTATORS
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Accounting for linear dependencies among – easy
– Schur stable
GUES:
Let
GUES
Extending to higher-order commutators – hard
DISCRETE TIME: BOUNDS on COMMUTATORS
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For they commute GUES, common Lyap fcn
– Schur stable
GUES:
Let
GUES
Example:
Our approach still GUES for
Perturbation analysis based on GUES for
CONTINUOUS TIME: LIE ALGEBRA STRUCTURE
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compact set of Hurwitz matrices
GUES: Lie algebra:
Levi decomposition: ( solvable, semisimple)
Switched transition matrix splits as where
and
Let and
GUESrobust condition butnot constructive
CONTINUOUS TIME: LIE ALGEBRA STRUCTURE
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more conservative buteasier to verifyGUES
There are also intermediate conditions
GUES:
Levi decomposition: ( solvable, semisimple)
Switched transition matrix splits as where
and
Let and
compact set of Hurwitz matrices
Lie algebra:
CONTINUOUS TIME: LIE ALGEBRA STRUCTURE
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Levi decomposition:
Switched transition matrix splits as
Previous slide: small GUES
But we also know: compact Lie algebra (not nec. small) GUES
Cartan decomposition: ( compact subalgebra)
Transition matrix further splits: where
and
Let
GUES
CONTINUOUS TIME: LIE ALGEBRA STRUCTURE
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Example:
GUES
Levi decomposition:
Cartan decomposition:
SUMMARY
Stability is preserved under arbitrary switching for:
• commuting subsystems:
• solvable Lie algebras (triangular up to coord. transf.)
• solvable + compact (purely imaginary eigenvalues)
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approximately
approximately
approximately