Finite-time boundedness of delayed Takagi-Sugeno quasi-one-sided Lipschitz systems

This paper investigates the finite-time boundedness (FBss) of perturbed nonlinear systems with interval time-varying delay. First, the system's nonlinear dynamics are represented by a Takagi–Sugeno (TS) fuzzy model, while quasi-one-sided Lipschitz (QL) terms account for either intrinsic nonlinearities or lumped uncertainties. This formulation captures a broad class of systems, encompassing both natural system nonlinearities and aggregated perturbations. Second, to ensure the FBss of the proposed system, an observer-based controller (OC) is employed under the assumption that the premise variables are measurable, even if the QL condition itself involves unmeasurable states. Then, a single-step procedure based on linear matrix inequalities (LMIs) is proposed to design the OC gains. To this end, a Lyapunov–Krasovskii (LK) functional is employed, comprising a delay-free quadratic term, which incorporates the matrix $\mathcal{P}$ used to verify the QL condition, along with three integral terms that capture the effects of the interval time-varying delay, including its lower and upper bounds. In addition, we employ a suitable decoupling approach that allows $\mathcal{P}$ to be regarded as a full decision-variable matrix and enabling the LMI conditions to be derived in a single step. The effectiveness and feasibility of the proposed methods are finally verified through a simulation study.