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) Lur'e 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 - 1 function G( s ) = + ( 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 Lur'e 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 .