An Extension of the Describing-Function Method

Abstract

The use of the describing-function method has been generally limited to systems with nonlinearities which can be combined in a block diagram of a feedback control system which is represented by a scalar variable. This paper deals with an extension of the method to some autonomous systems with more than one nonlinearity. The paper also shows that nonlinear systems can have hypersurfaces or manifolds in state space which divide regions of different stability characteristics. Each of these manifolds can be regarded as a boundary element of a family of surfaces related to a Lyapunov function. The describing-function method is applied in such a way that the hypersurfaces are approximated by a quadratic form. Two cases in an example, for which data obtained by analog computer and describing function approximations are compared, show that the amount of error caused by the approximation can be acceptably small, particularly in the subspace of the state space in which a system exhibits nonlinearities.