  In this paper, we will discuss how operational limitations affect
input-output behaviours of the system. In particular, we will provide
formulations for passivity and passivity indices of a nonlinear system given
operational limitations on the input and state variables. This formulation is
presented in the form of local passivity and indices. We will provide
optimisation based formulation to derive passivity properties of the system
through polynomial approximations. Two different approaches are taken to
approximate the nonlinear dynamics of a system through polynomial functions;
namely, Taylor's theorem and a multivariate generalisation of Bernstein
polynomials. For each approach, conditions for stability, dissipativity, and
passivity of a system, as well as methods to find its passivity indices, are
given. Two different methods are also presented to reduce the size of the
optimisation problem in Taylor's theorem approach. Examples are provided to
show the applicability of the results.
