Distributed Optimization for Fractional-Order Multi-Agent Systems Based on Adaptive Backstepping Dynamic Surface Control Technology

In this article, the distributed optimization problem is studied for a class of fractional-order nonlinear uncertain multi-agent systems (MASs) with unmeasured states. Each agent is represented through a system with unknown nonlinearities, unmeasurable states and a local objective function described by a quadratic polynomial function. A penalty function is constructed by a sum of local objective functions and integrating consensus conditions of the MASs. Radial basis function Neural-networks (RBFNNs) and Neural networks (NN) state observer are applied to approximate the unknown nonlinear dynamics and estimate unmeasured states, respectively. By combining the NN state observer and the penalty function, and the stability theory of the Lyapunov function, the distributed observer-based adaptive optimized backstepping dynamic surface control protocol is proposed to ensure the outputs of all agents asymptotically reach consensus to the optimal solution of the global objective function. Simulations demonstrate the effectiveness of the proposed control scheme.