Improved Circle and Popov Criteria for systems containing magnitude bounded nonlinearities

This paper presents improved versions of the Circle and Popov Criteria for Lure systems in which the nonlinear element is both sector and magnitude bounded. The main idea is to use the fact that if the nonlinearity is magnitude bounded and the linear system is asymptotically stable, then its state will be ultimately bounded. When the state enters this set of ultimate boundedness, it will satisfy a narrower sector condition which can then be used to prove stability in a wider set of cases than the standard Circle and Popov Criteria. The results are illustrated with some numerical examples.