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STABILITY under CONSTRAINED SWITCHING

STABILITY under CONSTRAINED SWITCHING. Daniel Liberzon. Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign. TWO BASIC PROBLEMS. Stability for arbitrary switching Stability for constrained switching.

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STABILITY under CONSTRAINED SWITCHING

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  1. STABILITY under CONSTRAINED SWITCHING Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign

  2. TWO BASIC PROBLEMS • Stability for arbitrary switching • Stability for constrained switching

  3. MULTIPLE LYAPUNOV FUNCTIONS – GAS – respective Lyapunov functions is GAS Useful for analysis of state-dependent switching

  4. MULTIPLE LYAPUNOV FUNCTIONS decreasing sequence GAS decreasing sequence [DeCarlo, Branicky]

  5. DWELL TIME The switching times satisfy – GES – respective Lyapunov functions dwell time

  6. DWELL TIME The switching times satisfy Need: – GES

  7. DWELL TIME The switching times satisfy – GES Need:

  8. DWELL TIME The switching times satisfy must be – GES Need:

  9. average dwell time – dwell time: cannot switch twice if # of switches on AVERAGE DWELL TIME

  10. average dwell time Theorem: [Hespanha ‘99]Switched system is GAS if Lyapunov functions s.t. • . # of switches on AVERAGE DWELL TIME Useful for analysis of hysteresis-based switching logics

  11. MULTIPLE WEAK LYAPUNOV FUNCTIONS • . Theorem: is GAS if observable for each s.t. there are infinitely many switching intervals of length For every pair of switching times s.t. have – milder than ADT Extends to nonlinear switched systems as before

  12. APPLICATION: FEEDBACK SYSTEMS (Popov criterion) linear system observable positive real Weak Lyapunov functions: See also invariance principles for switched systems in: [Lygeros et al., Bacciotti–Mazzi, Mancilla-Aguilar, Goebel–Sanfelice–Teel] Corollary: switched system is GAS if • s.t. infinitely many switching intervals of length • For every pair of switching times at • which we have

  13. STATE-DEPENDENT SWITCHING Switched system unstable for some no common But switched system is stable for (many) other switch on the axes is a Lyapunov function

  14. STATE-DEPENDENT SWITCHING Switch on y-axis level sets of level sets of GAS Switched system unstable for some no common But switched system is stable for (many) other

  15. STABILIZATION by SWITCHING – both unstable Assume: stable for some

  16. STABILIZATION by SWITCHING – both unstable Assume: stable for some So for each either or [Wicks et al. ’98]

  17. UNSTABLE CONVEX COMBINATIONS Can also use multiple Lyapunov functions Linear matrix inequalities

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