Fuzzy Fractional-Order Adaptive Control for Mobility-as-a-Service Orchestration: A Cybernetic Approach

Mobility-as-a-Service (MaaS) orchestration faces critical challenges in managing the Service Quality Index (SQI), Fleet Utilization Ratio (FUR), and Dynamic Pricing Index (DPI) under uncertain and time-varying urban demand. Existing approaches&#x2014;including multi-agent reinforcement learning (MARL) and hybrid learning-optimization methods&#x2014;lack provable stability guarantees and systematic uncertainty handling, limiting their reliability for safety-critical urban mobility systems. This paper introduces the first control-theoretic framework for MaaS that simultaneously provides fractional-order modeling, complete fuzzy rule coverage, and Lyapunov-guaranteed stability. The proposed Fractional-Order Adaptive Fuzzy Controller (FOAFC) embodies Ashby&#x2019;s Law of Requisite Variety through a dual mechanism: a proportional component that attenuates large-scale disturbances and provides global stability, and a complete 2,187-rule fuzzy inference system that amplifies fine-grained decision variety near the operating equilibrium&#x2014;together matching controller complexity to the inherent variety of the MaaS operational envelope. Fractional-order Gr&#x00FC;nwald-Letnikov dynamics with differentiation order <inline-formula> <tex-math notation="LaTeX">$q = 0.99$ </tex-math></inline-formula> are employed to explicitly capture the hereditary memory effects of urban traffic flow, including long-range congestion propagation and demand correlation timescales of approximately 30 seconds, which integer-order models structurally fail to represent. Lyapunov stability is rigorously established using the Fractional Barbalat&#x2019;s Lemma, providing asymptotic convergence guarantees despite parametric uncertainties and nonlinear system dynamics. Simulation results demonstrate 95.35% convergence to the target equilibrium&#x2014;representing an optimal engineering tradeoff between tracking accuracy and stability, as gains beyond the critical value <inline-formula> <tex-math notation="LaTeX">$K_{2} = 195$ </tex-math></inline-formula> cause complete system failure&#x2014;alongside 95.7% RMSE reduction over classical PID and triple dominance over pure adaptive control across RMSE, settling time (53.4 s, 4.3% faster), and control energy (20,402 units, 1.1% lower). Near-perfect operational compliance is achieved: 100% SQI adherence, 99.3% FUR efficiency, and 100% DPI stability. Monte Carlo validation across 30 independent trials under &#x00B1;20% initial condition variations and &#x00B1;15% parameter perturbations yields a 100% success rate and coefficient of variation of 3.47%, confirming robust, production-ready performance. This work provides a mathematically rigorous and operationally viable foundation for next-generation urban mobility systems.