The polytopic/fuzzy polynomial approach for non-linear control: advantages and drawbacks

A fairly general class of nonlinear plants can be modeled as as a time-varying polytopic combination of "vertex" linear systems also known as Takagi-Sugeno fuzzy systems. As many linear LMI control results naturally generalize to such systems, LMI formulations for TS control became the tool of choice in the 1990s. Important results have since been obtained, although significant sources of conservativeness remain. Using a Taylor-series formalism, a polytopic combination of polynomials can be used to describe such nonlinear systems. The sum-of-squares paradigm can be used for such polytopic polynomial systems. This paper reviews the main motivation behind such modelling techniques and the sources of conservatism of control designs based on the linear vertex models instead of the original nonlinear equations. The reader is referred to [31] for further discussion.