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

An improved sufficient condition for asymptotic stability developed for the monotone nonlinearity is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Re [Z(jω) (G(jω) + 1/k₂)]</tex> or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Re [Z(jω,) (G(jω) + 1/k₂)] ≥ 0</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z(s)=1++Y(s)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∫min{0}max{∞}|y(t)|dt<1</tex> . Examples of allowed <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z(s)</tex> functions are considered which show that the angle of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G(jω) +1/k₂</tex> is permitted to take on both positive and negative values outside the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">±90</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(T)</tex> in terms of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(O)</tex> . 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.