Download presentation

Presentation is loading. Please wait.

Published byHenry Jourdan Modified over 2 years ago

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, USA Yuliy Baryshnikov Bell Labs, Murray Hill, NJ, USA Andrei A. Agrachev S.I.S.S.A., Trieste, Italy 1 of 11

2
SWITCHED SYSTEMS Switched 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

3
STABILITY ISSUE unstable Asymptotic stability of each subsystem is not sufficient for stability under arbitrary switching 3 of 11

4
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

5
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

6
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 matrices there 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

7
DISCRETE TIME: BOUNDS on COMMUTATORS 7 of 11 – Schur stable Let GUES Idea of proof: take,, consider GUES:

8
DISCRETE TIME: BOUNDS on COMMUTATORS 7 of 11 – Schur stable GUES: Let GUES Idea of proof: take,, consider

9
DISCRETE TIME: BOUNDS on COMMUTATORS 7 of 11 induction basis contraction small – Schur stable GUES: Let GUES Idea of proof: take,, consider

10
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

11
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

12
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 GUES robust condition but not constructive

13
CONTINUOUS TIME: LIE ALGEBRA STRUCTURE 8 of 11 more conservative but easier to verify GUES 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:

14
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

15
CONTINUOUS TIME: LIE ALGEBRA STRUCTURE 10 of 11 Example: GUES Levi decomposition: Cartan decomposition:

16
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

Similar presentations

OK

1 Stability Analysis of Continuous- Time Switched Systems: A Variational Approach Michael Margaliot School of EE-Systems Tel Aviv University, Israel Joint.

1 Stability Analysis of Continuous- Time Switched Systems: A Variational Approach Michael Margaliot School of EE-Systems Tel Aviv University, Israel Joint.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on 555 timer ic Ppt on south african economy Ppt on soft skills for engineering students Ppt on herbs and spices Ppt on layer 3 switching vs layer Ppt on question tags examples Ppt on articles of association of a company Ppt on george bernard shaw Ppt on quality education group Ppt on asian continent clip