Revision:

1. Revision: What is the definition of K functions? What is the definition of KL functions?

2. Lecture 8 Stability of time-varying systems

3. Recommended reading • Khalil Chapter 3 (2nd edition)

4. Outline • Motivation • Tracking problems • Stability definitions • Stability via KL and K functions • Lyapunov conditions for UGAS/UGES • Summary

5. Motivation • Time-varying models are sometimes necessary to use when the properties of the system change significantly during the operation of the system. • Time-varying models may arise from tracking in time-invariant systems. • We consider stability properties of the origin. • Note: the definitions are more complicated than in the time-invariant case.

6. Tracking problems • Consider a time-invariant system and its reference solution Introduce x:=y-yr and write: • Origin for the x system corresponds to y(t)=yr(t) for the y system!

7. Stability definitions

8. Class of systems • We consider the following time-varying model where f is locally Lipschitz in x (uniform in t) and we assume that the origin is an equilibrium of the system:

9. Stability and uniform stability

10. The origin is stable if • The origin is uniformly stable if • The origin is unstable if it is not stable. • In stability  may depend on t0. • In uniform stability  is not allowed to depend on t0.

11. Stability Trajectory starting in (t0,) ball at time t1 may diverge! |x|  (t0,) (t1,) t0 t1 t

12. Example – non uniform stability • The system has the solution where s(t)=6 sin (t) – 6t cos(t)-t2 . For t0=2n we have that as n  then

13. Uniform stability |x| Uniform stability is a stronger property than stability!  () t0 t1 t

14. Attractivity and uniform attractivity

15. The origin is attractive if • The origin is uniformly attractive if • It is uniformly globally attractive if

16. Attractivity (special case c c(t0)) |x| We may need: T(t0,) < T(t1,) c T(t0,) T(t1,)  t T(t0,) t0 t1

17. Example – non uniform attractivity • The system has the solution The solution is attractive but not uniformly! The larger the t0, the longer it takes to converge.

18. Uniform attractivity |x| Exists T() independent of t0 c T()  t T() t0 t1

19. Asymptotic stability

20. The origin is asymptotically stable if it is stable and attractive. • The origin is uniformly asymptotically stable if uniformly stable and uniformly attractive. • It is uniformly globally asymptotically stable if it is uniformly stable and uniformly globally attractive.

21. We use the following acronyms: • UA  uniform attractivity • US  uniform stability • UGAS  uniform global asymptotic stability • ULAS  uniform local asymptotic stability • GAS  global asymptotic stability • LAS  local asymptotic stability

22. Comments • For time-invariant systems we have: • US  stability • UA  attractivity • ULAS  LAS • UGAS  GAS Hence, we often omit “U” in the acronyms!

23. Stability properties via K, KL functions

24. Why K and KL functions? • If the origin is US, there exist r>0, a K • If the origin is ULAS, there exist r>0,  KL • If the origin is UGAS there exists  KL

25. Lyapunov conditions

26. Theorem on UGAS • The origin of is UGAS if there exist V(t,x) and 1,2,3 K such that for all t and all x we have:

27. Sketch of proof – UGAS: • Along solutions we have: • Comparison principle: where u(t) is a solution to

28. Sketch of proof • We can assume that  is locally Lipschitz! This implies that the solution u(t)=(u0,t-t0) is a KL function. • This implies and finally

29. Theorem on UGES • The origin of is UGES if there exist V(t,x) and a1, a2, a3, c>0 such that for all t and all x we have:

30. Summary: • Time-varying systems may exhibit uniform or non-uniform stability properties. • K and KL functions are important tools in characterizing stability properties. • Lyapunov theorems for UGAS and UGES involve upper and lower bounds on V and on dV/dt that are independent of time t.

31. Next lecture: • Stability of perturbed systems. Homework: read Chapter 3 in Khalil

32. Thank you for your attention!