A Distributed Algorithm for Resource Allocation Over Dynamic Digraphs

A class of robust consensus algorithms with predefined‐time convergence under switching topologies

“…First, let us derive a useful lemma concerning average consensus in the absence of disturbance. Then, using an appropriate Lyapunov function, let us derive sufficient conditions for the design of control (7) such that the consensus is achieved in a predefined time under switching topologies Lemma 3. Let  (t) be a switching dynamic network composed of connected graphs and assume that under the protocol (7) and in the absence of disturbance, consensus is achieved at a time T c .…”

“…Then, according to Theorem 3, protocol (7) guarantees that the consensus is achieved before a predefined-time T c . Moreover, since the above argument holds for any  l ∈  , then protocol (7) guarantees that the consensus is achieved before a predefined-time T c , on switching dynamic networks under arbitrary switching. Furthermore, it follows from Lemma 3 that if Δ(t) = 0, then the consensus state x * is the average of the initial values of the agents.…”

“…The corresponding algebraic connectivity is 2 (( l )) ∈ {0.16548, 0.73648, 0.15776, 0.57104}. The initial conditions of the agents are randomly generated and are as follows: According to Theorem 6, protocol (7) with T c = 1, p = 1.5, q = 3.0, k = 0.5, = 1, = 2, { 1 , … , 4 } = {241.5668, 54.2777, 253.3879, 70.0029}, = 54.2777 and = 0.3693 guarantees that the consensus is achieved in a predefined-time T c under the switched dynamic network  (t) . Figure 4 shows the corresponding result.…”

“…In cooperative control, a fundamental challenge is the consensus problem, whose objective is to design decentralized protocols such that all the agents, in the network, interact with their neighbors to reach a common value, 1 called the consensus equilibrium. Many decentralized consensus-based methods have been applied, for instance, to flocking, 2 formation control, [3][4][5] distributed resource allocation, 6,7 and vehicle platooning. [8][9][10] Two consensus problems have been mainly addressed: the leader-follower consensus problem, 11,12 where the agents along the network converge to the state of the leader (real or virtual) that communicates its state only to a subset of agents; and the leaderless consensus problem, [13][14][15] where the consensus equilibrium is a function of the initial conditions of the agents, for instance, achieving consensus to the average value, 2 the average min-max value, 16,17 the median value, 18 and the maximum or minimum value 19 of the agents' initial conditions.…”

Distributed continuous‐time algorithm for a general nonsmooth monotropic optimization problem

“…For example, optimal resource allocation problems, which are known to be pretty useful in reality, have been addressed in the works of Beck et al and Ramírez‐Llanos and Martínez that are, in fact, a special case of monotropic optimization problems. Moreover, power systems have been taken into consideration in the works of Yi et al and Xu et al for the purpose of optimally allocating related power resources.…”