In this article, we consider the distributed optimal consensus problem under nominal and nonfragile cases for a class of minimum-phase uncertain nonlinear systems with unity-relative degree and disturbances generated by an external autonomous system. The involved cost function is the sum of all local cost functions associated with each individual agent. Two different edge-based distributed adaptive algorithms utilizing the internal model principle are designed to solve the problem in a fully distributed manner. Graph theory, nonsmooth analysis, convex analysis, and the Lyapunov theory are employed to show that the proposed algorithms converge accurately to the optimal solution of the considered problem. Finally, an example involving the dynamics of a Lorenz-type system is provided to demonstrate the effectiveness of the obtained results.

In this paper, based on differential inclusions and the saddle point dynamics, a novel second-order continuous-time algorithm is proposed to solve the optimal resource allocation problem in power systems. The considered cost function is the sum of all local cost functions with a set of affine equality demand constraints and an inequality constraint on generating capacity of the generator. In virtue of nonsmooth analysis, geometric graph theory, and Lyapunov stability theory, all generators achieve consensus on the Lagrange multipliers associated with a set of affine equality constraints while the proposed algorithm converges exponentially to the optimal solution of the resource allocation problem starting from any initial states over an undirected and connected graph. Moreover, the obtained results can be further extended to the optimal resource allocation problem in case of switching communication topologies. Finally, two numerical examples involving a smart grid system composed of five generators and the IEEE 30-bus system demonstrate the effectiveness and the performance of the theoretical results.