Consensus control of multi-agent systems is an innovative paradigm for the development of intelligent distributed systems. This has fascinated numerous scientific groups for their promising applications as they have the freedom to achieve their local and global goals and make their own decisions. Network communication topologies based on graph and matrix theory are widely used in a various real-time applications ranging from software agents to robotics. Therefore, while sustaining the significance of both directed and undirected graphs, this research emphases on the demonstration of a distributed average consensus algorithm. It uses the harmonic mean in the domain of multi-agent systems with directed and undirected graphs under static topologies based on a control input scheme. The proposed agreement protocol focuses on achieving a constant consensus on directional and undirected graphs using the exchange of information between neighbors to update their status values and to be able to calculate the total number of agents that contribute to the communication network at the same time. The proposed method is implemented for the identical networks that are considered under the directional and non-directional communication links. Two different scenarios are simulated and it is concluded that the undirected approach has an advantage over directed graph communication in terms of processing time and the total number of iterations required to achieve convergence. The same network parameters are introduced for both orientations of the communication graphs. In addition, the results of the simulation and the calculation of various matrices are provided at the end to validate the effectiveness of the proposed algorithm to achieve consensus.

Recent technological advancements in communication systems, control systems, and computer systems engineering attract many researchers to study the latest challenges and developments in the distributed control of multi-agent systems. It is a system with multiple agents, which are a self-managed and intelligent. The main objectives of such systems are to achieve their main objective in a cooperative manner that is essentially beyond the individual capabilities of a single agent. This is usually accomplished by properly designing a distributed protocol [

In this study, we propose a distributed algorithm based on average consensus and using the concept of harmonic mean to quickly converge to its optimal value. The proposed algorithm is applied to both directed and undirected graphs for fixed networks in order to visualize their meaning for real word problems. The rest of the article is organized as follows. Section II provides a preliminary study on the design of a consensus algorithm. Section III deals with the formulation of the proposed consensus algorithm problem. The results of the simulation are presented in Section IV, and finally Section V concludes the article.

We use state space functions to model the proposed network control system. In cooperative control of multi-agent systems, graph theory and matrix theory are used as the main tools to study and design such systems together with the state space model.

In mathematics and in particular in graph theory, a graph is a representation of a set of objects in which certain pairs of objects are connected by links. Connected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges. The edges can be directed or undirected. The vertices that are directly connected to the other vertices of a network are considered its neighbors and are represented mathematically as

If the graph does not have a specific direction for the edges, it is called an undirected graph. In undirected graph, the communication links for the information flow are bidirectional. For undirected graphs, ^{T}, so the matrix is symmetric with respect to the diagonal. An ordered pair is used to denote the edge on a graph and can be written as

If the edges are directional, it is a directed graph. The edges are connected from node to node, and some of the same paths cannot be used to return to the original node. In directional diagrams, arrows on the edges indicate the direction of information flow. The source node can only use the specified address to reach the destination node, but must find an alternate route to return the start node. In some cases, a directed graph has problems related to the point of no return. This can lead to a blockage of communication networks. Furthermore, it can be said that a directed graph is a sequence of directed edges in the form of

In addition to graph theory, convergence modeling for networked multi-agent systems depends mainly on matrix theory. When all the entries in a matrix are positive, the matrix is said to be a non-negative matrix. Similarly, with respect to the shape of the vector, the vector is not negative if all elements are positive. In matrix theory, different matrices are of great importance for convergence analysis. One of them is a stochastic matrix (row)

One of the most important matrix for designing a consensus algorithms is the degree matrix. The Degree matrix

Consider a multi-agent system consisting of _{i} for _{i}. The agent should consider local state information as well as information from other connected agents. We investigate the following spatial state model in this investigation in

In _{i} is related to the control input generated by the agent. We have to design it so that all agents have to converge towards a common value based on their average. Furthermore, agent _{i}, which changes dynamically with time iteration. Furthermore, _{i} _{i}

The design of a control input _{i}

In

Here, _{i} and _{j} are the degrees of agent

This section deals with the numerical and simulation results generated with the MATLAB simulation tool for the proposed algorithm. In this section we demonstrate the convergence analysis of our algorithm and also calculate the total number of agents in a network. A root mean square error is also recorded to ensure minimum error and achieve precision of results. Two cases are presented to reinforce our claim that the proposed algorithm achieves consensus with good results.

The following parameters need to be defined at the beginning of each simulation scenario.

Total number of agents (Network Size)

Initial conditions for all agents in a network

Computed value of weighting factor

Network topology (Adjacency): Fixed, directed, undirected

In the current scenario, we simulate it for directional and undirected orientations of the diagram. It also calculated the various matrices and provided a comparison for the performance of both graphs in the given network environment.

We consider a small undirected graphics network topology with

We can formulate and compute the Adjacency matrix A, Degree matrix D, and the Laplacian matrix L for the undirected communication graph considered above as follows:

SIA Matrix =

We consider a small undirected graphics network topology with

We can formulate and compute the Adjacency matrix A, Degree matrix D, and the Laplacian matrix L for the undirected communication graph considered above as follows:

SIA Matrix =

The convergence achieved in relation to several iterations and the time required for directional and undirected diagrams are shown in

Undirected Graph | Directed Graph | |
---|---|---|

Number of Iterations | 8 | 23 |

Computational Time | 2.5 | 10 |

In the current scenario, we simulate it for directional and undirected orientations of the diagram. It also calculated the various matrices and provided a comparison for the performance of both graphs in the given network environment with eight agents.

First, we consider a multi-agent network with _{1} _{i}

We can formulate the Adjacency, Degree, and the Laplacian matrix for the undirected graph considered above as follows:

SIA Matrix =

We consider a multi-agent network with _{1} _{i}

We can formulate the Adjacency, Degree, and the Laplacian matrix for the undirected graph considered above as follows:

The convergence achieved in relation to several iterations and the time required for directional and undirected cases in scenarios are shown in

Undirected Graph | Directed Graph | |
---|---|---|

Number of Iterations | 24 | 53 |

Computational Time | 20 | 50 |

This article presents an implementation of the proposed algorithm in directed and undirected graphs for the cooperative control of multi-agent systems. To determine the total number of agents in a network, the proposed algorithm converges with the reciprocal of the total number of agents. This implementation is done with stochastic matrix tools and algebraic graph theories. We simulate two cases of directional and non-directional alignment of the diagram to check the effectiveness of the proposed model for the multiple landscapes. Both objectives set for the research are achieved by attaining the convergence and the finding the agent count. The simulation results show that the proposed algorithm is very efficient in case of directed graphs with respect to a series of iterations and calculation time as compared for undirected graphs. One of the reasons is the bidirectional information flow, which can be easily evaluated on the undirected diagram. This work can be extended in the future by simulating the same network configuration in different network environments, e.g., adding noise and uneven delays. In addition, the update time of the agent states can be updated in an asynchronous clock mode and time-limited convergence. It can be further extended by adding more key performance indicators such as convergence time and the convergence factor to evaluate the convergence analysis of the proposed model.