1 / 14

A LIE-ALGEBRAIC CONDITION for STABILITY of SWITCHED NONLINEAR SYSTEMS

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

sherry
Download Presentation

A LIE-ALGEBRAIC CONDITION for STABILITY of SWITCHED NONLINEAR SYSTEMS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


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

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

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

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

  7. SWITCHED LINEAR SYSTEMS Lie algebra w.r.t. Assuming GES of all modes, GUES is guaranteed for: • 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) Quadratic common Lyapunov function exists in all these cases Extension based only on L.A. is not possible [Agrachev & L ’01]

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

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

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

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

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

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

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

More Related