TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS

1. TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS Andrei A. Agrachev S.I.S.S.A., Trieste, Italy Yuliy Baryshnikov Bell Labs, Murray Hill, NJ, USA Daniel Liberzon Univ. of Illinois, Urbana-Champaign, USA 49th CDC, Atlanta, GA, Dec 2010 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 Discrete dynamics classes of switching signals Properties of the continuous state Details of discrete behavior are “abstracted away” : stability 2 of 11

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

4. KNOWN RESULTS: LINEAR SYSTEMS • commuting subsystems: • nilpotent Lie algebras (suff. high-order Lie brackets are 0) • solvable Lie algebras (triangular up to coord. transf.) e.g. • solvable + compact (purely imaginary eigenvalues) [Narendra–Balakrishnan, Agrachev–L] Gurvits, Kutepov, L–Hespanha–Morse, Lie algebra w.r.t. Stability is preserved under arbitrary switching for: Quadratic common Lyapunov function exists in all these cases No further extension based on Lie algebra only 4 of 11

5. KNOWN RESULTS: NONLINEAR SYSTEMS GUAS • Linearization (Lyapunov’s indirect method) • Global results beyond commuting case Recently obtained using optimal control approach [Margaliot–L ’06, Sharon–Margaliot ’07] • Commuting systems Can prove by trajectory analysis[Mancilla-Aguilar ’00] or common Lyapunov function[Shim et al. ’98, Vu–L ’05] 5 of 11

6. Checkable conditions • In terms of the original data • Independent of representation • Not robust to small perturbations In anyneighborhood of any pair of matrices there exists a pair of matrices generating the entire Lie algebra [Agrachev–L ’01] REMARKS on LIE-ALGEBRAIC CRITERIA How to capture closeness to a “nice” Lie algebra? 6 of 11

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

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

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

10. DISCRETE TIME: BOUNDS on COMMUTATORS GUES Accounting for linear dependencies among – easy – Schur stable GUES: Let Extending to higher-order commutators – hard 7 of 11

11. DISCRETE TIME: BOUNDS on COMMUTATORS GUES Example: For they commute GUES, common Lyap fcn Our approach still GUES for Perturbation analysis based on GUES for – Schur stable GUES: Let 7 of 11

12. CONTINUOUS TIME: LIE ALGEBRA STRUCTURE 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 8 of 11

13. CONTINUOUS TIME: LIE ALGEBRA STRUCTURE compact set of Hurwitz matrices Lie algebra: Levi decomposition: ( solvable, semisimple) Switched transition matrix splits as where and GUES GUES: Let and more conservative but easier to verify There are also intermediate conditions 8 of 11

14. CONTINUOUS TIME: LIE ALGEBRA STRUCTURE But we also know: compact Lie algebra (not nec. small) GUES Cartan decomposition: ( compact subalgebra) Transition matrix further splits: where and Let GUES Levi decomposition: Switched transition matrix splits as Previous slide: small GUES 9 of 11

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

16. SUMMARY Stability is preserved under arbitrary switching for: approximately • commuting subsystems: approximately • solvable Lie algebras (triangular up to coord. transf.) approximately • solvable + compact (purely imaginary eigenvalues) 11 of 11