This paper investigates the finite-time mixed \(H_{\infty }\) and passive control problem for singular Caputo fractional-order systems (SCFOSs) with polytopic uncertainties. Addressing the absence of finite-time results for singular fractional-order models under combined energy-based and dissipativity requirements, we develop a unified framework that achieves a mixed \(H_{\infty }\) -passivity performance index within a single tunable structure. By integrating finite-time stability theory with the nonlocal properties of the Caputo derivative, new admissibility conditions are established to ensure regularity, impulse-freeness, and finite-time boundedness of the closed-loop SCFOS. To enhance numerical scalability, a strict LMI-based output-feedback design is proposed through a novel variable transformation, which removes equality constraints and significantly improves computational tractability for systems with large uncertainty polytopes. The resulting controller guarantees robust finite-time performance with the unified mixed \(H_{\infty }\) –passivity index for all admissible uncertainty realizations. Two numerical examples, including a fractional-order electrical circuit, are provided to demonstrate the correctness, reduced conservatism, and computational efficiency of the proposed method.
