STABILITY under CONSTRAINED SWITCHING Daniel Liberzon Coordinated Science
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
MULTIPLE LYAPUNOV FUNCTIONS – GAS – respective Lyapunov functions is GAS Useful for analysis of state-dependent switching
MULTIPLE LYAPUNOV FUNCTIONS decreasing sequence GAS decreasing sequence [De. Carlo, Branicky]
DWELL TIME The switching times satisfy – GES – respective Lyapunov functions dwell time
DWELL TIME The switching times satisfy – GES Need:
DWELL TIME The switching times satisfy – GES Need:
DWELL TIME The switching times satisfy – GES Need: must be
AVERAGE DWELL TIME # of switches on average dwell time – dwell time: cannot switch twice if
AVERAGE DWELL TIME # of switches on average dwell time Theorem: [Hespanha ‘ 99] Switched system is GAS if Lyapunov functions s. t. • • Useful for analysis of hysteresis-based switching logics
MULTIPLE WEAK LYAPUNOV FUNCTIONS Theorem: • is GAS if . observable for each • • s. t. there are infinitely many switching intervals of length – milder than ADT • For every pair of switching times s. t. have Extends to nonlinear switched systems as before
APPLICATION: FEEDBACK SYSTEMS (Popov criterion) linear system observable positive real Weak Lyapunov functions: Corollary: switched system is GAS if • s. t. infinitely many switching intervals of length • For every pair of switching times which at we have See also invariance principles for switched systems in: [Lygeros et al. , Bacciotti–Mazzi, Mancilla-Aguilar, Goebel–Sanfelice–Teel]
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
STATE-DEPENDENT SWITCHING Switched system unstable for some no common But switched system is stable for (many) other Switch on y-axis level sets of GAS
STABILIZATION by SWITCHING – both unstable Assume: stable for some
STABILIZATION by SWITCHING – both unstable Assume: stable for some So for each either or [Wicks et al. ’ 98]
UNSTABLE CONVEX COMBINATIONS Can also use multiple Lyapunov functions Linear matrix inequalities
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