Local stability of composite systems--Frequency-domain condition and estimate of the domain of attraction

This paper is concerned with such composite systems whose subsystems contain one nonlinearity each and whose interconnections are functions of the scalar outputs of subsystems. A frequency-domain condition which assures local asymptotic stability is given under the assumptions that each nonlinearity satisfies a sector condition, that interconnections are linearly bounded, and that linear parts of subsystems may have unstable poles. In deriving the above result, such Lyapunov functions of subsystems are constructed so that their weighted sum is a Lyapunov function of the overall system. A method to estimate the domain Of attraction based on the above Lyapunov functions is also studied. When the bounds on nonlinearities hold true in the entire space and when the linear parts do not have unstable poles, the present condition turns out to be the same with the L
2
-stability condition which was obtained before by Araki.