LMI relaxations for nonlinear fuzzy control systems via homogeneous polynomials

Based on recent results on homogeneous polynomially parameter-dependent (HPPD) solutions to parameter-dependent LMIs (PD-LMIs) that arise from robust stability of linear parameter varying (LPV) systems, we investigate the relaxed conditions characterized by parameter-dependent LMIs (PD-LMIs) in terms of firing strength belonging to the unit simplex, exploiting the algebraic property of Polyapsilas theorem to construct a family of finite-dimensional LMI relaxations. The main contribution of this paper is that sets of relaxed LMIs are parameterized in term of the polynomial degree d. As d increases, progressively less conservative LMI conditions are generated, being easier satisfied due to more freedom provided by new variables involved. An example to illustrate the relaxation is provided.