Absolute Stability of Second-Order Systems With Asymmetric Sector Boundaries

Abstract

Absolute stability is considered under an asymmetric sector condition, resulting in a generalization of the classical absolute stability analysis. In addition to the extension to asymmetric sector conditions, the current approach, based on the recent work of Leonov (2005), yields results that are both necessary and sufficient for absolute stability. An asymmetric sector condition allows different sector bounds to be imposed in different regions of the state space, and removes the restriction that the system nonlinearity be confined to only the first and third quadrants. This work applies to second-order systems. The necessary and sufficient conditions are derived by comparing the vector field of the nonlinear system with that of certain piecewise linear systems. As an example, stabilization of a supercavitating (underwater) vehicle system is considered and it is shown that the new results are less conservative than those obtained with classical theory, which requires imposition of a symmetric sector condition.