Stability Analysis and Generalized Memory Controller Design for Delayed T–S Fuzzy Systems via Flexible Polynomial-Based Functions

In this article, stability analysis and controller synthesis problems for Takagi–Sugeno (T–S) fuzzy systems with time-varying delay are studied. A generalized parameter-dependent reciprocally convex inequality (GPDRCI) is presented to handle the derivative of triple integral terms, which is more general than some existing ones. By choosing suitable flexible polynomials with tunable parameters, novel flexible polynomial-based functions (FPFs) are proposed in delay-product types, which overcome the incompletely slack matrices, higher-order time delay and insufficient parameters in the existing functions. Benefitting from completely slack matrices and lower-order time delay, coupling relationship among system states and time delay is fully linked. Based on the GPDRCI and FPFs, a stability condition is derived for T–S fuzzy systems. Based on the stability criterion, considering both the time-varying delay and its bounds, a generalized memory controller is designed for T–S fuzzy systems, which covers the memoryless and traditional memory ones as its special cases. In addition, the constraints on introduced slack matrices in some existing works are avoided with the help of a matrix inequality decoupling technique. These provide extra free dimensions in the solution space. Some examples are employed to illustrate the effectiveness of the proposed methods.