Converse Lyapunov Theorem Research Articles

A smooth Lyapunov function from a class-${\mathcal{KL}}$ estimate involving two positive semidefinite functions

We consider dierential inclusions where a positive semidenite function of the solutions satises a class-KL estimate in terms of time and a second positive semidenite function of the initial condition. We show that a smooth converse Lyapunov function, i.e., one whose derivative along solutions can be used to establish the class-KL estimate, exists if and only if the class-KL estimate is robust, i.e., it holds for a larger, perturbed dierential inclusion. It remains an open question whether all class-KL estimates are robust. One sucient condition for robustness is that the original dierential inclusion is locally Lipschitz. Another sucient condition is that the two positive semidenite functions agree and a backward completability condition holds. These special cases unify and generalize many results on converse Lyapunov theorems for dierential equations and dierential inclusions that have appeared in the literature.

Dynamical systems which undergo switching

The authors investigate the stability of a system in which the dynamics at any instant in time will follow one of a fixed set of vector fields. They allow switching between members of the family of vector fields to be completely random. The focus of this paper is to prove a converse Lyapunov theorem for this class of systems.

New converse Lyapunov theorems and related results on exponential stability

We treat the problem of constructing Lyapunov functions for systems which are, by assumption, exponentially stable. The construction we present results in a larger set of functions than those obtainable by previously known methods. A useful property of the proposed Lyapunov functions is that they preserve information on the rate of exponential convergence of the system. Some useful applications are given.

An adaptive observer for robots with persistent excitation

An adaptive state observer for rigid link manipulators is proposed. The proposed adaptive state observer is based on parametrizing the robot dynamic model via the estimated state and parameters, thus yielding a nominal model plus two perturbation terms, one is linear and the other is quadratic in the state and parameter estimation errors. The update law for the adaptive observer is derived using the MKY lemma (Narendra and Annaswamy 1989). Stability of the origin of the overall error system is shown using the Gronwall-Belmann lemma (Khalil 1992) and a converse Lyapunov theorem (Khalil 1992). The proposed adaptive observer is tested through simulations

A Converse Lyapunov Theorem with Applications to Iss-Disturbance Attenuation

This work provides a generalization of a converse Lyapunov theorem for systems that do not have the regularity property on the invariant sets. As an application, we provide Lyapunov-function like necessary and sufficient characterizations for the problem of ISS-disturbance attenuation.

Stabilization with Respect to Noncompact Sets: Lyapunov Characterizations and Effect of Bounded Inputs

Abstract This paper deals with the global smooth stabilization of nonlinear systems with respect to not necessarily compact sets. We prove a converse Lyapunov theorem, and present a result on stability under input perturbations.