Input-output stability of Lur´e systems with inputs satisfying bounding conditions on magnitude and slope

This paper considers the stability of Lur´e systems in the sense that the outputs are bounded for any input having bounded two norms, or bounded infinity norms, on both magnitude and slope, where the linear subsystem belongs to a large subclass of convolution systems. It is shown that if the well-known Popov criterion is satisfied, then the system is stable in the above sense for any nonlinearity lying in a sector bound. Based on the Popov criterion, a practical inequality for determining stability points by numerical methods is developed. To illustrate the usefulness of the contribution of the paper, a numerical example is provided, in which the plant is a nonrational transfer function.