Generalised absolute stability and Sum of Squares

This paper introduces a general framework for analysing systems that have non-polynomial, uncertain or high order nonlinearities. It decomposes the vector field using Lur´e type feedback into a system with a polynomial or rational vector field and a nonlinear memoryless feedback term, which is bounded by polynomial or rational functions. This decomposition can be used to model uncertainty in the nonlinear term or to bound difficult to analyse nonlinearities by simpler polynomial or rational functions. Conditions for stability are found using Lyapunov functions which are generalisations of those used for the derivation of the multivariable circle and Popov criteria. These conditions can be given in terms of polynomial inequalities and so Sum of Squares techniques can be used to efficiently analyse these systems. An example shows how the techniques can be applied to uncertain coupled genetic circuits and a pendulum, where the nonlinearity is bounded by polynomial functions. The technique is also applied to show global stability of a system in which classical absolute stability is inconclusive.