Distributed time-varying constrained optimisation for multi-agent systems: theory and application

In this paper, a distributed time-varying constrained optimisation problem is investigated for multi-agent system. A specified-time practical consensus approach is presented to deal with the considered problem. First, distributed continuous-time algorithms with consensus part and optimal part are proposed. In consensus part, practical specified-time convergence algorithm coupled with a Hessian-dependent gain is presented. In optimal part, a penalty-based method is provided to address the nonlinear inequality constraints. Then, by designing the Hessian-dependent gain, the effects of the nonuniform Hessian matrix of the penalised cost function can be eliminated. Based on this algorithm, the state consensus agreement is realised within a desired time, and the system can realise the asymptotical tracking of the optimal solution. Thirdly, the specified-time convergence and the asymptotic optimality of the system are proved using Lyapunov arguments. Finally, the effectiveness of the proposed algorithms is demonstrated by the results of numerical simulation and experiment.