Stability and Stabilization With Additive Freedom for Delayed Takagi–Sugeno Fuzzy Systems by Intermediary-Polynomial-Based Functions

This paper is devoted to the stability and stabilization for Takagi-Sugeno fuzzy systems with time-varying delays. First, an improved matrix inequality is presented to bound both strictly and nonstrictly proper rational functions, which is more general than the existing versions of reciprocally convex lemmas. Second, by suitable operations on parameter-dependent polynomial multiplied by state rate, a couple of novel intermediary-polynomial-based functions (IPFs) are developed in delay-product types. Benefitting from slack matrices of IPFs, a certain degree of flexibility is furnished. More importantly than that, by feat of adjustment of the variable parameter, the resulting conditions will be further endowed with additive freedom, which relaxes the feasible space in a distinctive manner. Third, by utilizing IPFs along with triple integrals, the stability criteria and the controller design approach are derived by some advanced integral inequalities. Resorting to elaborate construction of IPFs, the strengths of bounding techniques are sufficiently exploited, and the information on delay derivative is adequately reflected. Consequently, more desirable performances are achieved, while without excessive computational complexity. Finally, the effectiveness of the proposed methods is verified by numerical examples.