Absolute Stability Criteria for Systems with Sector or Norm Bounded Nonlinearities and Uncertain Delay

Finding conditions for absolute stability of a system containing a linear part and a scalar nonlinear sector restricted function is a classical Lur´e problem. Most of the corresponding results are based on the frequency domain or Lyapunov functions methods which are applied to systems with a time-invariant or periodic linear block. This paper develops a new approach to stability analysis of the problem based on a direct analysis of the corresponding integral Volterra equation about the input of the nonlinear block. The obtained sufficient stability criterion is applicable to non-autonomous systems with arbitrary time-varying delay in the feedback. The approach is extended to general time-varying systems including a linear block and norm bounded vector nonlinear terms with uncertain time-varying delays. The obtained delay-independent stability conditions are formulated in the terms of the transition matrix of the linear part and the norms of the nonlinear terms. The systems are indicated for which the obtained criteria are not only sufficient but also necessary for any delay function. The obtained results are applied to stability analysis of some systems previously studied in the literature; in all cases less conservative stability bounds are found.