An improved frequency time domain stability criterion for autonomous continuous systems

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Abstract
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 functionG(j\omega), isRe(1 + X(j\omega) + Y(j\omega) + \alphaj\omega)(G(j\omega) + 1/k_{2}) \geq 0where\alpha>0 , x(t)\leq 0fort \leq 0andx(t)=0fort>0 , y(t)\leq0fort>0andy(t) = 0fort < 0, and\int\min{-\infty}\max{\infty}(| x(t)| + | y(t) | )dt < 1. The improvement consists of the addition of theX(j\omega)term which corresponds to a nonzero time function fort<0, resulting inZ(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.