Optimal strictly positive real approximations for stable transfer functions

The problem of finding the optimal strictly positive real (SPR) approximation to a given stable transfer function is considered. The transfer function is further assumed to be strictly proper and the SPR approximation is constrained to have the same pole structure. The optimisation is carried out using the (weighted) H₂-norm and the problem is reduced to a strictly convex quadratic programming problem with linear inequality constraints. At the heart of the method is a parametrisation for all SPR compensators which possess a given denominator polynomial. Motivation for the problem stems from the robust stability provided by SPR compensation for passive plants such as flexible structures with collocated sensing and actuation. Numerical examples are provided, as well as the experimental implementation of an optimal approximation to the control of a single-flexible-link manipulator