Synthesis of absolutely stabilising controllers

The synthesis of fixed order controllers for nonlinear systems with sector bounded nonlinearities is considered. The authors constructed an inner and outer approximation of the set of absolutely stabilising linear controllers by casting the closed loop system as a Lure-Postnikov system. The inner approximation is based on the well-known sufficient conditions that require strict positive realness (SPR) of open loop transfer function (possibly with some multipliers) and a characterisation of SPR transfer functions that require a family of complex polynomials to be Hurwitz. The outer approximation is based on the condition that the open loop transfer function must have infinite gain margin, which translates to a family of real polynomials being Hurwitz. The authors illustrate the proposed methodology through the construction of an inner and outer approximation of absolutely stabilising controllers for a mechanical system.