Robust synchronization of Caputo–Hadamard fractional-order T–S fuzzy Lur'e systems with delay: LMI and SOS conditions

In this paper, the robust asymptotic synchronization conditions are obtained for uncertain nonlinear Caputo–Hadamard fractional-order Takagi–Sugeno (T–S) fuzzy Lur'e-type systems with time-varying delay. Matched fuzzy-rule weights represent the master and slave plants, and the slave system is controlled by a static output-error parallel distributed compensation controller that can utilize either the current or past output errors. Norm-bounded structures for the uncertain matrices and incremental sector bounds on the local nonlinearities are considered. Delay-free and delayed synchronization criteria are obtained through the use of a common quadratic Lyapunov function, a Caputo–Hadamard quadratic derivative inequality, the Schur complement, and a Caputo–Hadamard Halanay inequality. The constant-matrix results are further extended to polynomial T–S fuzzy Lur'e systems using a global unconstrained sum-of-squares matrix certificate. For a real-life partial-output scenario (with fewer measured outputs than state dimension), the numerical validation first solves the delayed linear matrix inequalities (LMI) as a max-margin semidefinite program using CVXPY, and then describes the polynomial extension via an explicit matrix sum-of-squares (SOS) Gram certificate and an additional simulation example. The main novelty is that, to the best of my knowledge, this is the first study to address simultaneously, within a single convex LMI/SOS framework, the Caputo–Hadamard (logarithmic-time) fractional order, an unbounded time-varying delay, matched norm-bounded parametric uncertainty, incremental sector nonlinearities, and static partial output-error feedback; unlike integer-order or Riemann–Liouville formulations, this setting cannot rely on the classical chain rule and instead requires a Caputo–Hadamard quadratic derivative inequality together with a logarithmic-time Halanay argument. Throughout, "robust synchronization" is meant in the sense of complete synchronization (asymptotic decay of the error to zero) that is guaranteed for every admissible uncertainty; a numerical study further quantifies how the fractional order $\alpha$ shapes the synchronization transient.