TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta,...
-
Upload
henry-jourdan -
Category
Documents
-
view
212 -
download
0
Transcript of TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta,...
![Page 1: TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,](https://reader035.fdocuments.in/reader035/viewer/2022070308/551c2c2f5503469e4f8b60ef/html5/thumbnails/1.jpg)
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
1 of 11
![Page 2: TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,](https://reader035.fdocuments.in/reader035/viewer/2022070308/551c2c2f5503469e4f8b60ef/html5/thumbnails/2.jpg)
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
2 of 11
![Page 3: TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,](https://reader035.fdocuments.in/reader035/viewer/2022070308/551c2c2f5503469e4f8b60ef/html5/thumbnails/3.jpg)
STABILITY ISSUE
unstable
Asymptotic stability of each subsystem is
not sufficient for stability under arbitrary switching
3 of 11
![Page 4: TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,](https://reader035.fdocuments.in/reader035/viewer/2022070308/551c2c2f5503469e4f8b60ef/html5/thumbnails/4.jpg)
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
![Page 5: TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,](https://reader035.fdocuments.in/reader035/viewer/2022070308/551c2c2f5503469e4f8b60ef/html5/thumbnails/5.jpg)
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
5 of 11
![Page 6: TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,](https://reader035.fdocuments.in/reader035/viewer/2022070308/551c2c2f5503469e4f8b60ef/html5/thumbnails/6.jpg)
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?
6 of 11
![Page 7: TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,](https://reader035.fdocuments.in/reader035/viewer/2022070308/551c2c2f5503469e4f8b60ef/html5/thumbnails/7.jpg)
DISCRETE TIME: BOUNDS on COMMUTATORS
7 of 11
– Schur stable
Let
GUES
Idea of proof: take , , consider
GUES:
![Page 8: TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,](https://reader035.fdocuments.in/reader035/viewer/2022070308/551c2c2f5503469e4f8b60ef/html5/thumbnails/8.jpg)
DISCRETE TIME: BOUNDS on COMMUTATORS
7 of 11
– Schur stable
GUES:
Let
GUES
Idea of proof: take , , consider
![Page 9: TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,](https://reader035.fdocuments.in/reader035/viewer/2022070308/551c2c2f5503469e4f8b60ef/html5/thumbnails/9.jpg)
DISCRETE TIME: BOUNDS on COMMUTATORS
7 of 11
inductionbasis
contraction small
– Schur stable
GUES:
Let
GUES
Idea of proof: take , , consider
![Page 10: TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,](https://reader035.fdocuments.in/reader035/viewer/2022070308/551c2c2f5503469e4f8b60ef/html5/thumbnails/10.jpg)
DISCRETE TIME: BOUNDS on COMMUTATORS
7 of 11
Accounting for linear dependencies among – easy
– Schur stable
GUES:
Let
GUES
Extending to higher-order commutators – hard
![Page 11: TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,](https://reader035.fdocuments.in/reader035/viewer/2022070308/551c2c2f5503469e4f8b60ef/html5/thumbnails/11.jpg)
DISCRETE TIME: BOUNDS on COMMUTATORS
7 of 11
For they commute GUES, common Lyap fcn
– Schur stable
GUES:
Let
GUES
Example:
Our approach still GUES for
Perturbation analysis based on GUES for
![Page 12: TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,](https://reader035.fdocuments.in/reader035/viewer/2022070308/551c2c2f5503469e4f8b60ef/html5/thumbnails/12.jpg)
CONTINUOUS TIME: LIE ALGEBRA STRUCTURE
8 of 11
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
![Page 13: TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,](https://reader035.fdocuments.in/reader035/viewer/2022070308/551c2c2f5503469e4f8b60ef/html5/thumbnails/13.jpg)
CONTINUOUS TIME: LIE ALGEBRA STRUCTURE
8 of 11
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:
![Page 14: TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,](https://reader035.fdocuments.in/reader035/viewer/2022070308/551c2c2f5503469e4f8b60ef/html5/thumbnails/14.jpg)
CONTINUOUS TIME: LIE ALGEBRA STRUCTURE
9 of 11
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
![Page 15: TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,](https://reader035.fdocuments.in/reader035/viewer/2022070308/551c2c2f5503469e4f8b60ef/html5/thumbnails/15.jpg)
CONTINUOUS TIME: LIE ALGEBRA STRUCTURE
10 of 11
Example:
GUES
Levi decomposition:
Cartan decomposition:
![Page 16: TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,](https://reader035.fdocuments.in/reader035/viewer/2022070308/551c2c2f5503469e4f8b60ef/html5/thumbnails/16.jpg)
SUMMARY
Stability is preserved under arbitrary switching for:
• commuting subsystems:
• solvable Lie algebras (triangular up to coord. transf.)
• solvable + compact (purely imaginary eigenvalues)
11 of 11
approximately
approximately
approximately