Semi-global practical asymptotic stability and averaging

Stability of periodic solutions for Lipschitz systems obtained via the averaging method

“…Since nonsmooth systems appear frequently in applications (simple operations as the max-operator or existence of constraints lead inevitably to a loss of differentiability) it is important to study the corresponding stability problem. In [15], it has been assumed that the nonsmooth system (1.1) admits the constant solution x ≡ z 0 and that F is (Lipschitz on z and) independent of ε. In that case, a study of stability of the constant solution has been carried out, via the global asymptotic stability of the averaged system y = εf (y); see [15,Theorem 2].…”

“…In [15], it has been assumed that the nonsmooth system (1.1) admits the constant solution x ≡ z 0 and that F is (Lipschitz on z and) independent of ε. In that case, a study of stability of the constant solution has been carried out, via the global asymptotic stability of the averaged system y = εf (y); see [15,Theorem 2]. In [10, Theorem 3.1] it has been proved that the same result holds when F is continuous and degree zero homogeneous.…”

On Non-Local Stability Properties of Extremum Seeking Control

“…Finally, we emphasize that our proof technique is novel and is based on Lyapunov techniques and recent new developments in the theory of averaging (Nešić and Teel, 2001;Teel, 2000;A. R. Teel and Aeyels, 1999) and singular perturbations (Christofides and Teel, 1996;Teel et al, 2003) that are tailored for analysis of semi-global practical stability of systems that exhibit time scale separation.…”

Stability analysis via averaging for nonlinear systems with delays

“…A powerful tool in this context is the averaging method, see, for instance, [1], [3], [5], [6], [7], [10], [11], and references therein. The key idea of the averaging method is to convert the stability analysis for a time varying system to the analysis of a time invariant system that can be approximated in certain manners by the original system as the small parameter approaches 0 (see Definitions 3.1 and 3.4).…”

“…The main difference of the two types of averaged systems lies in the roles played by the external inputs in the approximation. For a system not affected by external inputs, the strong-and weak-averaged systems both become averaged systems used for the original system, see [11].…”