Practical Fixed-Time Distributed Optimization for Uncertain High-Order Nonlinear Multi-Agent Systems under Mismatched Disturbances

This paper investigates the practical fixed-time distributed optimization problem (DOP) in nonlinear multi-agent networks, where the optimization decision of each agent is subject to global consensus and local set-inclusion constraints. Unlike existing distributed optimization results, this work considers high-order nonlinear multi-agent systems with unknown nonlinearities and mismatched disturbances. To address the challenges arising from the complex high-order structures, a novel disturbance-rejection distributed optimization control framework is proposed, which integrates fixed-time control theory based on the sign function and consensus-based optimality conditions. The proposed algorithm is implemented through an active disturbance rejection backstepping design procedure, consisting of two key stages: first, a fixed-time extended state observer is developed to estimate the unknown nonlinear terms and mismatched disturbances in the system; second, a distributed fixed-time optimization controller is constructed based on fixed-time state and gradient observers, which eliminates the need for an optimal signal generator and thus avoids additional tracking errors. Furthermore, rigorous Lyapunov stability analysis guarantees the practical fixed-time stability of the proposed method, namely, the system error converges to a bounded neighborhood of the origin within a fixed time. Finally, numerical simulations validate the effectiveness and robustness of the proposed framework.