Jurdjevic-Quinn Conditions

Abstract

AbstractThe Jurdjevic-Quinn Theorem provides a powerful framework for guaranteeing globally asymptotic stability, using a smooth feedback of arbitrarily small amplitude. It requires certain algebraic conditions on the Lie derivatives of a suitable non-strict Lyapunov function, in the directions of the vector fields that define the system. The non-strictness of the Lyapunov function is an obstacle to proving robustness, since robustness analysis typically requires strict Lyapunov functions. In this chapter, we provide a method for overcoming this obstacle. It involves transforming the non-strict Lyapunov function into an explicit global CLF. This gives a strict Lyapunov function construction for closed-loop Jurdjevic-Quinn systems with feedbacks of arbitrarily small magnitude. This is valuable because (a) the non-strict Lyapunov function from the Jurdjevic-Quinn Theorem is often known explicitly and (b) our methods apply to Hamiltonian systems, which commonly arise in mechanical engineering. We illustrate our work using a two-link manipulator model, as well as an integral input-to-state stability result.