Remark on stability margin of Lur'e systems

In a recent work, a definition of stability margin (gain and phase margins) for a class of nonlinear systems (Lur'e systems consisting of a linear time-invariant (LTI) plant and a sector-constrained nonlinearity) is proposed based on the famous circle criterion. This definition is indeed interesting because the concept of gain and phase margins has been largely limited to linear systems with a single input and a single output (SISO), but it was further established for Lur'e systems with a particular type of sector constraints ( $k_1=0$, e.g., saturation). In this paper, the previously established concept is extended to cover Lur'e systems with a general type of sector constraints ( $k_1\ge 0$). It is, however, pointed out that in case of $k_1=0$, the definitions of phase margin based on the circle criterion can be misleading and need to be modified or replaced perhaps by a time-delay margin for Lur'e systems including an integrator, for which the phase margin can be trivially zero.