Robust adaptive fixed-time distributed optimization for second-order nonlinear multi-agent systems under input saturation

We propose a fixed-time robust adaptive tracking control strategy for the fixed-time distributed optimization problem of a second-order nonlinear multi-agent system (MAS) with unbalanced graphs, input saturation, matched and unmatched disturbances. First, a fixed-time distributed optimization reference signal based on gradient-sum estimation is designed to realize the fixed-time distributed optimization of an unbalanced reference signal network. Second, a fixed-time disturbance estimation algorithm is designed for mismatched disturbances. The hierarchical coordinated tracking error is defined based on disturbance estimation and reference signals, and the second-order nonlinear MAS is transformed into a dynamic tracking error system. The unknown and continuously nonlinear part is approximated by an FLS (fuzzy logic system). Third, using the fixed-time sliding mode control method, a sliding mode dynamic system model with a composite learning robust adaptive law is established, and the robust adaptive term of the system is estimated online to compensate for the unknown disturbance terms. Then, the sliding mode variable is embedded in the smooth hyperbolic tangent function to obtain a fixed-time robust adaptive tracking algorithm. The algorithm further realizes the fixed-time distributed optimization of the second-order nonlinear MAS. Finally, two simulation examples verify the algorithm's good performance.