Noncausal multipliers for nonlinear system stability

Using the Popov approach, new absolute stability conditions in multiplier form are derived for a single-loop system with a time-invariant stable linear element
G
in the forward path and a nonlinear time-varying gain
k(t)\phi(\dot)
in the feedback path. The classes of nonlinearities considered are the monotonic, odd monotonic, and power law. The stability multiplier contains causal and noncausal terms; for absolute stability, the latter give rise to a lower bound (which is believed to be new) on
dk/dt
and the former, as in earlier investigations, to an upper bound on
dk/dt
. Asymptotic stability conditions for a linear system are realized as a limiting case of the absolute stability conditions derived for the power law nonlinearity.