Nonsymmetric Laplacian Matrix Research Articles

Distributed Cooperative Learning for Discrete-Time Strict-Feedback Multi Agent Systems Over Directed Graphs

This paper focuses on the distributed cooperative learning (DCL) problem for a class of discrete-time strict-feedback multi-agent systems under directed graphs. Compared with the previous DCL works based on undirected graphs, two main challenges lie in that the Laplacian matrix of directed graphs is nonsymmetric, and the derived weight error systems exist n-step delays. Two novel lemmas are developed in this paper to show the exponential convergence for two kinds of linear time-varying (LTV) systems with different phenomena including the nonsymmetric Laplacian matrix and time delays. Subsequently, an adaptive neural network (NN) control scheme is proposed by establishing a directed communication graph along with n-step delays weight updating law. Then, by using two novel lemmas on the extended exponential convergence of LTV systems, estimated NN weights of all agents are verified to exponentially converge to small neighbourhoods of their common optimal values if directed communication graphs are strongly connected and balanced. The stored NN weights are reused to structure learning controllers for the improved control performance of similar control tasks by the “mod” function and proper time series. A simulation comparison is shown to demonstrate the validity of the proposed DCL method.

Asymptotical rotating consensus algorithm with processing delay

In this paper, we studied some consensus algorithms for the collective rotating motions of a team of agents, which has been widely studied in different disciplines ranging from physics, networks and engineering. Both discrete and continues consensus algorithm with processing delays are investigated. There are three motion patterns determined by the information exchange topology of systems and rotation angle of rotation matrices. The asymptotic consensus appears when 0 is an simple eigenvalue of Laplacian matrix and the rotation angle is less than the critical value, and the rotating consensus achieves when the rotation angle is equal to the critical value. At this point, all agents move on circular orbits and the relative radii of orbits are equal to the relative magnitudes of the components of a right eigenvector associated with 0 eigenvalue of the non-symmetric Laplacian matrix. Finally, all agents move along logarithmic spiral curves with a fixed center when the rotation angle is larger than the critical value.

Consensus of Second-order Multi-agent Systems with Directed Networks Using Relative Position Measurements Only

This brief paper studies the consensus problem of second-order multi-agent systems when the agents’ velocity measurements are unavailable. Firstly, two simple consensus protocols which do not need velocity measurements of the agents are derived to guarantee that the multi-agent systems achieve consensus in directed networks. Secondly, a key constant which is determined by the complex eigenvalue of the nonsymmetric Laplacian matrix and an explicit expression of the consensus state are respectively developed based on matrix theory. The obtained results show that all the agents can reach consensus if the feedback parameter is bigger than the key constant. Thirdly, the theoretical analysis shows that the followers can track the position and velocity of the leader provided that the leader has a directed path to all other followers and the feedback parameter is bigger enough. Finally, numerical simulations are given to illustrate the effectiveness of the proposed protocols.

Periodic coordinated rotating motion control of second-Order multi-Agent systems under directed interaction topologies

ABSTRACTThis paper investigates the problem of periodic motion control of second-order multi-agent systems in three dimensions under directed interaction topologies. Distributed algorithms for periodic rotating motion around a static and moving target were proposed by exploring the introduction of Cartestian coordinate coupling for each agent. In case of a static target, we show that when the nonsymmetric Laplacian matrix has certain properties, the damping gain is above a certain bound, and the Euler angle is equal to a critical value, all the agents will eventually rotate around the target periodically. In case of a moving target, the composite motion behaviours with translation and rotation will emerge, when this moving target's information can only be available to one or one subset of these agents and all agents have only local interactions. Tools like matrix theory, linear system theory and other mathematical skills were used for convergence analysis. Simulation results were provided to illustrate the effectiveness of the theoretical results.

Modeling of physical network systems

Conservation laws and balance equations for physical network systems typically can be described with the aid of the incidence matrix of a directed graph, and an associated symmetric Laplacian matrix. Some basic examples are discussed, and the extension to k-complexes is indicated. Physical distribution networks often involve a non-symmetric Laplacian matrix. It is shown how, in case the connected components of the graph are strongly connected, such systems can be converted into a form with balanced Laplacian matrix by constructive use of Kirchhoff’s Matrix Tree theorem, giving rise to a port-Hamiltonian description. Application to the dual case of asymmetric consensus algorithms is given. Finally it is shown how the minimal storage function for physical network systems with controlled flows can be explicitly computed.

Sampled-data collective rotating consensus for second-order networks under directed interaction

This paper investigates collective rotating motions of second-order multi-agent systems in 3-D under a sampled-data setting. A rotating consensus protocol was proposed and conditions on sampling period, damping gain, communication topology and rotating angle such that the vehicles will eventually move on a straight-line path, cylindrical spirals and logarithmic, respectively, were derived. In particular, when the vehicles move along circular orbits, the relative radii of the orbits (respectively, the relative phases of the vehicles on their orbits) are equal to the relative magnitudes (respectively, the relative phases) of the components of a right eigenvector associated with the critical eigenvalue of the nonsymmetric Laplacian matrix. Simulations are performed to validate the theoretical results.

Convergence of rotating consensus algorithm for second-order dynamics in three dimensions

This paper investigates the collective rotating motions of a group of agents with second-order dynamics in three dimensions. A rotating consensus algorithm for such agents was proposed by introducing a Cartesian Coordinate Coupling and a velocity consensus item to an existing consensus protocol. This note shows that the resulting rotating motions can be affected by network topology, the damping gain and the Euler angle. It is also shown that convergence of the rotating consensus can be achieved when the damping gain is above a certain bound, the Euler angle is below a critical value and the communication network topology has a directed spanning tree. Furthermore, we show that the agents will eventually keep moving on a straight-line path, cylindrical spirals, and logarithmic columnar curves when the damping gain is above a certain bound, the Euler angle is below, equal and above a critical value, respectively. In particular, when the agents eventually move on circular orbits, the relative radii of the orbits (respectively, the relative phases of the agents on their orbits) are equal to the relative magnitudes (respectively, the relative phases) of the components of a right eigenvector associated with a critical eigenvalue of the nonsymmetric Laplacian matrix. Simulation results are presented to demonstrate the proposed algorithm.

Non–Symmetric Finite Networks: The Two–Point Resistance

Abstract An explicit formula for the resistance between two nodes in a network described by non-symmetric Laplacian matrix L is obtained. This is of great advantage eg in electronic circuit fault analysis, where non-linear systems have to be solved repeatedly. Analysis time can be greatly reduced by utilization of the obtained formula. The presented approach is based on the “mutual orthogonality” of the full system of left and right-hand eigenvectors of a diagonalizable matrix L. Simple examples are given to demonstrate the accuracy of this approach to circuit networks

Collective Motion From Consensus With Cartesian Coordinate Coupling

Collective motions including rendezvous, circular patterns, and logarithmic spiral patterns can be achieved by introducing Cartesian coordinate coupling to existing consensus algorithms. We study the collective motions of a team of vehicles in 3D by introducing a rotation matrix to an existing consensus algorithm for double-integrator dynamics. It is shown that the network topology, the damping gain, and the value of the Euler angle all affect the resulting collective motions. We show that when the nonsymmetric Laplacian matrix has certain properties, the damping gain is above a certain bound, and the Euler angle is below, equal, or above a critical value, the vehicles will eventually rendezvous, move on circular orbits, or follow logarithmic spiral curves lying on a plane perpendicular to the Euler axis. In particular, when the vehicles eventually move on circular orbits, the relative radii of the orbits (respectively, the relative phases of the vehicles on their orbits) are equal to the relative magnitudes (respectively, the relative phases) of the components of a right eigenvector associated with a critical eigenvalue of the nonsymmetric Laplacian matrix. Simulation results are presented to demonstrate the theoretical results.