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 Useful for analysis of state-dependent switching – GAS – respective Lyapunov functions is GAS

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

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

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

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

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

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

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

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

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

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

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

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

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

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