A LIEALGEBRAIC CONDITION for STABILITY of SWITCHED NONLINEAR
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: • • is a family of systems is a switching signal 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 Properties of the continuous state: stability
STABILITY ISSUE unstable Asymptotic stability of each subsystem is not sufficient for stability
TWO BASIC PROBLEMS • Stability for arbitrary switching • Stability for constrained switching
TWO BASIC PROBLEMS • Stability for arbitrary switching • Stability for constrained switching
GLOBAL UNIFORM ASYMPTOTIC STABILITY GUAS is Lyapunov stability plus asymptotic convergence GUES:
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) e. g. • solvable Lie algebras (triangular up to coord. transf. ) • 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]
SWITCHED NONLINEAR SYSTEMS • Commuting systems => GUAS [Mancilla-Aguilar, Shim et al. , Vu & L] • Linearization (Lyapunov’s indirect method) • Global results beyond commuting case – ? ? ? [Unsolved Problems in Math. Systems and Control Theory]
SPECIAL CASE globally asymptotically stable Want to show: Will show: differential inclusion is GAS is GUAS
OPTIMAL CONTROL APPROACH Associated control system: where (original switched system ) Worst-case control law [Pyatnitskiy, Rapoport, Boscain, Margaliot]: fix and small enough
MAXIMUM PRINCIPLE (along optimal trajectory) Optimal control: is linear in (unless at most 1 switch GAS )
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
GENERAL CASE Want: polynomial of degree (proof – by induction on bang-bang with switches GAS )
THEOREM Suppose: • GAS, backward complete, analytic • s. t. and Then differential inclusion is GAS (and switched system is GUAS)
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