Stability of systems with power-law non-linearities

Abstract

It is shown that a sufficient condition for the asymptotic stability-in-the-large of an autonomous system containing a linear part with transfer function G( jω) and a non-linearity belonging to a class of power-law non-linearities with slope restriction [0, K] in cascade in a negative feedback loop is ReZ(jω)[G(jω) + 1 K ] ≥ 0 for all ω where the multiplier is given by, Z(jω) = 1 + αjω + Y(jω) − Y(−jω) with a real, y( t) = 0 for t < 0 and ∫ 0 ∞ |y(t)| dt < 1 2 c 2 , c 2 being a constant associated with the class of non-linearity. Any allowable multiplier can be converted to the above form and this form leads to lesser restrictions on the parameters in many cases. Criteria for the case of odd monotonic non-linearities and of linear gains are obtained as limiting cases of the criterion developed. A striking feature of the present result is that in the linear case it reduces to the necessary and sufficient conditions corresponding to the Nyquist criterion. An inequality of the type |R(T) − R(− T)| ≤ 2c 2R(0) where R( T) is the input-output cross-correlation function of the non-linearity, is used in deriving the results.