LMI Relaxations for Quadratic Stabilization of Guaranteed Cost Control of T–S Fuzzy Systems

Less conservative condition is provided in this paper for quadratic stabilization of guaranteed cost control (GCC) of Takagi–Sugeno fuzzy systems with parallel distributed compensation (PDC). To derive the condition, firstly a parameter-dependent linear matrix inequality (PD-LMI) is established to find quadratically stable PDC controller gains of GCC. Secondly, by applying Pólya’s theorem, evaluation of the PD-LMI is transformed into an equivalent problem of evaluation of a sequence of LMI relaxations. Different from other existing conditions, the LMI relaxations are sufficient and asymptotically reach necessity for evaluating the PD-LMI as a related scalar parameter, d, increases. The resulting guaranteed costs of PDC controllers are non-increasing with respect to the increase in the parameter d and converge to the global optimal value under quadratic stability at the limiting case. In addition, for input-affine nonlinear systems, the proposed condition is extended with the consideration of modeling errors, which helps to reduce the computational complexity of the LMI relaxations. Finally, simulations of two examples demonstrate the efficiency and feasibility of the proposed condition.