Novel limit cycle estimation of solar free-piston Stirling engines via sampled-data T–S fuzzy method

Abstract This paper presents a sampled-data Takagi–Sugeno (T–S) fuzzy approach for estimating the limit cycle of free-piston Stirling engines (FPSEs). FPSEs are promising systems for clean and efficient power generation; however, their startup behavior, stable oscillation, and parameter design remain challenging because of their inherent nonlinear dynamics and the absence of mechanical linkages between the moving components. Unlike Mamdani-type fuzzy formulations, which rely on heuristic rule bases and often lack a rigorous stability framework, the proposed T–S fuzzy method offers three distinct advantages: (i) it enables systematic Lyapunov-based stability analysis through linear rule consequents; (ii) it provides a natural setting for robustness evaluation against parameter uncertainties; and (iii) it significantly reduces computational complexity by avoiding iterative rule-tuning procedures. To further enhance the stability conditions, a novel two-sided delay-dependent looped-functional is introduced, which relaxes the monotonicity limitations of conventional Lyapunov–Krasovskii functionals and better exploits the actual sampling pattern. A mathematical model of the FPSE is developed and used to implement the proposed framework. The method is validated through numerical simulations and comparison with available experimental data. The results demonstrate that the proposed approach estimates the limit cycle and tracks the motion of both the displacer and power pistons with a maximum relative error below 3%, confirming its accuracy and practical utility . Overall, the proposed T–S fuzzy sampled-data framework provides an effective, computationally tractable, and theoretically rigorous tool for the analysis of FPSE dynamics under the considered conditions.