Relative Stability Via the Direct Method of Lyapunov

Abstract

A mathematically rigorous concept of relative stability based on the v-functions of the direct method of Lyapunov is introduced. Two systems of the type representable by ẋ = f (x ) are considered, where under the proper restrictions on f (x ), a Lyapunov function, v(x ) is uniquely determined by a positive definite error criterion r(x ) and the equation v̇(x ) = −r(x ). The definition of the relative stability proposed, eventually leads to conditions on the linear approximation systems which are sufficient to assure the relative stability of the nonlinear systems. This leads to conditions on the eigenvalues of the linear approximation system which are necessary but not sufficient for relative stability. Additional conditions on the choice of the error criteria are needed. The present definition permits the gap between concepts of stability in classical control theory and that due to the direct method of Lyapunov to be at least partially bridged.