Input-to-state stability and boundedness in Lurie systems

We analyze the input-to-state stability (ISS) of the feedback interconnection of a linear block and a nonlinear element. This study is of importance for establishing robustness against actuator nonlinearities and disturbances. In the absolute stability framework, we prove ISS from a positive real property of the linear block, by restricting the sector nonlinearity to grow unbounded as its argument tends to infinity. When this growth condition is violated, examples show that the ISS property is lost. The result is used to give a simple proof of boundedness for negative resistance oscillators, such as the van der Pol oscillator. In a separate application, we relax the minimum phase assumption of an earlier boundedness result for systems with nonlinearities that grow faster than linear.