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TOWARDS ROBUST LIE-ALGEBRAIC STABILITY
CONDITIONS for SWITCHED LINEAR SYSTEMS
Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
European Control Conference, Budapest, Aug 2009
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
STABILITY ISSUE
unstable
Asymptotic stability of each subsystem is
not sufficient for stability under arbitrary switching
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
Assuming GES of all modes, GUES is guaranteed 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]
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
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 measure closeness to a “nice” Lie algebra?
EXAMPLE
(discrete time, or cont. time periodic switching: )
Suppose .
Fact 1 Stable for dwell time :
Fact 2 If then always stable:
This generalizes Fact 1 (didn’t need ) and Fact 2 ( )
Switching between and
Schur
Fact 3 Stable if where is small enough s.t.
When we have where
EXAMPLE
(discrete time, or cont. time periodic switching: )
Suppose .
Fact 1 Stable for dwell time :
Fact 2 If then always stable:
Switching between and
Schur
When we have where
MORE GENERAL FORMULATION
Assume switching period :
MORE GENERAL FORMULATION
Find smallest s.t.
Assume switching period
MORE GENERAL FORMULATION
Assume switching period
Intuitively, captures:
• how far and are from commuting ( )
• how big is compared to ( )
, define byFind smallest s.t.
MORE GENERAL FORMULATION
Stability condition: where
already discussed: (“elementary shuffling”)
Assume switching period
, define byFind smallest s.t.
SOME OPEN ISSUES
• Relation between and Lie brackets of and
general formula seems to be lacking
SOME OPEN ISSUES
• Relation between and Lie brackets of and
Suppose but
Elementary shuffling: ,
not nec. close to but commutes with and
• Smallness of higher-order Lie brackets
Example:
SOME OPEN ISSUES
• Relation between and Lie brackets of and
Suppose but
Elementary shuffling: ,
not nec. close to but commutes with and
This shows stability for
Example:
In general, for can define by
• Smallness of higher-order Lie brackets
(Gurvits: true for any )
SOME OPEN ISSUES
• Relation between and Lie brackets of and
• Smallness of higher-order Lie brackets
• More than 2 subsystems
can still define but relation with Lie brackets less clear
• Optimal way to shuffle
especially when is not a multiple of
• Current work investigates alternative approaches
that go beyond periodic switching
