A combined frequency-time domain stability criterion for autonomous continuous systems

An improved sufficient condition for asymptotic stability developed for the monotone nonlinearity is
Re [Z(j\omega) (G(j\omega) + 1/k_{2})]
or
Re [Z(j\omega,) \bar(G(j\omega) + 1/k_{2})] \geq 0
where
Z(s)=1+\alphas+Y(s)
and
\int\min{0}\max{\infty}|y(t)|dt<1
. Examples of allowed
Z(s)
functions are considered which show that the angle of
G(j\omega) +1/k_{2}
is permitted to take on both positive and negative values outside the
\pm90\deg
band. The approach used in deriving the above results is a general one which reduces the problem of obtaining sufficient conditions for asymptotic stability to one of obtaining a bound for a cross-correlation function
R(T)
in terms of
R(O)
. As an example, a sufficient condition for asymptotic stability is developed for a continuous nonlinearity having a monotone bound which appears to have considerable practical application.