Absolute stability analysis through a connection to saturation nonlinearities

A generalized sector was introduced recently for improved stability analysis of systems with nonlinearity and/or uncertainty. While providing more flexibility and admitting more accuracy in the description of the nonlinear/uncertain component, the generalized sector is almost as numerically tractable as the traditional conic sector - necessary and sufficient conditions for absolute quadratic stability were identified in the form of linear matrix inequalities (LMIs) for continuous-time systems with one nonlinear component. The objective of this paper is to develop a general framework for absolute stability analysis of systems with multiple nonlinear components under a generalized sector condition. Through a connection between saturation functions and piecewise linear convex/concave functions, the generalized sector is described in terms of a set of saturation functions. This transforms the problem of absolute stability analysis into one of stability analysis for systems with saturation nonlinearities, for which effective tools have recently been developed. Under the general framework, we develop explicit conditions for absolute quadratic stability of discrete-time systems with one nonlinear component.