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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 :.

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A LIE-ALGEBRAIC CONDITION for STABILITY of SWITCHED NONLINEAR SYSTEMS

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  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)

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