A stability inequality for a class of nonlinear feedback systems

For some systems, the Popov stability criterion fails to verify Aizerman's conjecture, that is, when the Popov sector is not equal to the linear (Hurwitz) sector. In these cases, the question of stability for a nonlinearity which exceeds the Popov sector, but which is included in the Hurwitz sector, is unanswered. This paper provides a partial answer to this question by taking into account the slope of the nonlinear function. By constraining this slope to the interval
[-k_{1}, k_{2}]
and the nonlinearity to the sector
[0, k]
, the following stability inequality is obtained
Re (1 + j\omegaq)G(j\omega) + 1/k +\mu\omega^{2}{1 + (k_{2}-k_{1})Re G(j\omega)-k_{1}k_{2}| G(j\omega)|^{2}} >0
where μ is a non-negative parameter. For
\mu=0
this inequality reduces to the Popov criterion. Two examples are given, in the first of which the sector is extended up to the linear limit. The Popov theorem concerned only the zero-input response of the nonlinear feedback system~ whereas here a restricted class of inputs to the system is allowed.