Synchronization Of Multi-agent Systems With Matrix-Weighted Coupling

Cooperative control of multi-agent systems has attracted much attention in the field of systems and control science,and has great application potential in unmanned system clusters,sensor networks and future autonomous combat systems.The consensus of multi-agent systems is the most basic and important problem in cooperative control.In this thesis,the synchronization of multi-agent systems is studied based on the matrix-weighted coupling.The main feature of matrix-weighted coupling multi-agent systems is that matrix weight is used to represent the coupling or dependence of multi-dimensional states among agents or individuals.The introduction of matrix-weighted coupling expands the scalar-weighted coupling case,but also increases the difficulty of system stability analysis and control synthesis.Matrix-weighted coupled dynamical systems have gradually appeared in LC oscillator system array and the sociological opinion evolution model.In recent years,matrix-weighted coupled dynamical systems have become a hotspot of synchronization and cluster analysis in control field.Although some theories and methods have been applied to the synchronization analysis and controller design of matrix-weighted coupling systems,there are still some deficiencies in these theories and methods.For example,the spectral property analysis of the matrix-weighted Laplacians is only applicable to the case of symmetric coupling,when the coupling matrix is asymmetric,the spectral property research has not obtained results.The synchronization of multi-agent systems with scalar-weighted coupling is often studied by using non-negative matrix theory or the maximum and minimum state difference functions of agents as candidate Lyapunov functions to analyze the stability,but these techniques are no longer applicable to multi-agent systems with matrixweighted coupling.In this thesis,the spectral properties of Laplacian matrices,synchronization of the time-invariant coupling multi-agent systems,the design of the controllers and synchronization of the time-varying coupling multi-agent systems are studied.The main work includes the following aspects:Firstly,for a class of time-invariant second-order oscillator systems with matrix-weighted coupling,synchronization control protocols are presented by sampling data when the state derivative information of the system is not available.Necessary and sufficient conditions depending on the coupling gains,the sampling period,and the spectra of the matrix-weighted Laplacian,are established for achieving synchronization in the case of undirected graph.Based on the assumption of spectral properties of matrix-weighted Laplacian,the necessary and sufficient conditions for synchronization in the case of digraph are obtained when the coupling is asymmetric,and a class of topological graphs satisfying the assumption of spectral properties are constructed.Secondly,distributed synchronization protocols are designed for multi-agent systems with time-varying output coupling.Under bidirectional coupling,the synchronization conditions of the multi-agent systems are established by using the output information.Under unidirectional coupling,based on the assumption of detectability and appropriate graph connectivity conditions,a distributed synchronization protocol is designed by making use of switched system theory,and the corresponding convergence results are established.Then,the synchronization problem of a class of multi-agent systems with time-varying matrix-weighted coupling is studied.Based on persistent graphs and joint connectivity conditions,the sufficient conditions for exponential synchronization of multi-agent systems are established with the help of adaptive control theory.The results extend the multi-agent singleintegrator consensus model with scalar-weighted coupling in the existing literature,and are applied to multi-agent systems with time-varying and non-identical coupling matrices,which extend the asymptotic synchronization results in literature to exponential synchronization.Finally,we study the spectral properties of signed Laplacian matrices of a special class of undirected matrix-weighted graphs,which are signed graphs with scalar coupling of positive and negative weights.By means of matrix tree theorem,it is proved that when p negative edges exist in undirected signed graph,the corresponding signed Laplacian matrix contains at most p negative eigenvalues;and for the signed Laplacian corresponding to a tree,it has exactly p negative eigenvalues.An algebraic sufficient and necessary condition for the positive semi-definite of the signed Laplacians with a simple zero eigenvalue is given.Furthermore,we establish the upper and lower bounds for the second smallest eigenvalue of the signed Laplacians when adding a negative weighted edge to the signed graphs,and show that these bounds are tight.