1. A LIE-ALGEBRAIC CONDITION for STABILITY of SWITCHED NONLINEAR SYSTEMS CDC ’04 Michael Margaliot Tel Aviv University, Israel Daniel Liberzon Univ. of Illinois at Urbana-Champaign, USA

2. SWITCHED vs. HYBRID SYSTEMS : stability Switching can be: State-dependent or time-dependent Autonomous or controlled Hybrid systems give rise to classes of switching signals Properties of the continuous state Switched system: is a family of systems is a switching signal Further abstraction/relaxation: diff. inclusion, measurable switching

3. STABILITY ISSUE unstable Asymptotic stability of each subsystem is not sufficient for stability

4. TWO BASIC PROBLEMS Stability for arbitrary switching Stability for constrained switching

5. TWO BASIC PROBLEMS Stability for arbitrary switching Stability for constrained switching

6. GLOBAL UNIFORM ASYMPTOTIC STABILITY GUAS is Lyapunov stability plus asymptotic convergence GUES:

7. SWITCHED LINEAR SYSTEMS Extension based only on L.A. is not possible [Agrachev & L ’01] Lie algebra w.r.t. Quadratic common Lyapunov function exists in all these cases Assuming GES of all modes, GUES is guaranteed for: commuting subsystems: nilpotent Lie algebras (suff. high-order Lie brackets are 0) e.g. solvable Lie algebras (triangular up to coord. transf.) solvable + compact (purely imaginary eigenvalues)

8. SWITCHED NONLINEAR SYSTEMS Global results beyond commuting case – ??? Commuting systems Linearization (Lyapunov’s indirect method) => GUAS [Mancilla-Aguilar, Shim et al., Vu & L] [Unsolved Problems in Math. Systems and Control Theory]

9. SPECIAL CASE globally asymptotically stable Want to show: is GUAS Will show: differential inclusion is GAS

10. OPTIMAL CONTROL APPROACH Associated control system: where (original switched system ) Worst-case control law [Pyatnitskiy, Rapoport, Boscain, Margaliot] : fix and small enough

11. MAXIMUM PRINCIPLE is linear in at most 1 switch (unless ) GAS Optimal control: (along optimal trajectory)

12. SINGULARITY Need: nonzero on ideal generated by (strong extremality) At most 2 switches GAS Know: nonzero on strongly extremal (time-optimal control for auxiliary system in ) constant control (e.g., ) Sussmann ’79:

13. GENERAL CASE GAS Want: polynomial of degree (proof – by induction on ) bang-bang with switches

14. THEOREM Suppose: GAS, backward complete, analytic s.t. and Then differential inclusion is GAS (and switched system is GUAS)