A revisited Popov criterion for nonlinear Lur'e systems with sector-restrictions

This paper revisits a well-known Popov criterion for absolute stability analysis of multiple sector-restricted nonlinear time-invariant (NTI) Lure systems. Extending the Brockett and Willems (1965) frequency-domain Popov criterion for a SISO system into a MIMO system with multiple sector-restrictions 0, , where is positive and diagonal, provides a claim that a system is absolutely stable if a function G(s) = (-1) + (I + Ms)G(s) is strictly positive real, where G(s) is the transfer function from uncertain outputs to uncertain inputs and M is now diagonal and real. However, a Lyapunov Lure function has been found only for a non-negative diagonal M but not for a real diagonal M, which makes researchers confine M as non-negative diagonal. However, in this paper, we show that this Lyapunov function is still valid for a real diagonal matrix M.