An improved frequency time domain stability criterion for autonomous continuous systems

A sufficient condtion given for the asymptotic stability of a system having a single monotonic nonlinearity with slope confined to
[0, k_{2}]
and a transfer function
G(j\omega)
, is
Re(1 + X(j\omega) + Y(j\omega) + \alphaj\omega)(G(j\omega) + 1/k_{2}) \geq 0
where
\alpha>0 , x(t)\leq 0
for
t \leq 0
and
x(t)=0
for
t>0 , y(t)\leq0
for
t>0
and
y(t) = 0
for
t < 0
, and
\int\min{-\infty}\max{\infty}(| x(t)| + | y(t) | )dt < 1
. The improvement consists of the addition of the
X(j\omega)
term which corresponds to a nonzero time function for
t<0
, resulting in
Z(j\omega)
multipliers whose phase angle is capable of varying from +90° to -90° any desired number of times. As is shown by examples, the new criterion gives better results than existing criteria. Also developed is an improved criterion for an odd monotonic nonlinearity.