Edge Consensus Problems For Complex Networks With Nonnegative Constraints

As an important problem in the field of cooperative control of complex networks,consensus problems have aroused many attentions from reasearches in many areas.This thesis studies the edge consensus problem of complex networks,the theoretical system of line graph provides the foundation to analyze edge networks,the construction of edge network and the relation between the given nodal network and edge network will bring some new ideas to consensus problems of multi-agent systems.Moreover,the positivity constraints of edge states will bring difficulities to the theoretical analysis,some new theoretical should be proposed.First,it proposes the edge network based on a given nodal network,and gives the edge dynamics model.The relations between the given nodal network and the edge network are presented,and addition with the theoretical and technology of positive systems,the difficulities caused by positivity properties are first solved.Furthermore,the edge consensus problems with positivity constraints and nonlinear dynamics are studied.It enriches the theoretical system of multi-agent system.The main achievements are given as follows:For the edge consensus problem based on state feedback method,the positive edge dynamics are presented,and the nonnegative edge consensus results are presented for both undirected and directed nodal networks whose edges are described by nominal or uncertain continuous-time positive linear systems.The definition and topology of line graph are first derived from the given nodal graph to represent the interaction of the edges.The lower and upper bounds of the nonzero eigenvalues and the general algebraic connectivity of the Laplacian matrix for the line graph are given based on the connections between nodal network and edge network.According to the properties of Metzler matrix,sufficient conditions for consensus with positivity constraints are given with the combination effects of the edge system matrices,the vertex number,and the edge number of the original nodal network.Finally,the feedback matrix is given by solving the ARE equation or the linear programming problem.For the nonnegative edge consensus problem via output feedback method,twoobserver-based consensus algorithms are introduced,the selection of appropriate augmented variable is discussed in detail to satisfy the positivity constraints.Necessary and sufficient conditions for nonnegative edge consensus are proposed.Furthremore,without utilizing the global information of topology information,sufficient conditions for nonnegative edge consensus with nominal or uncertain dynamics are constructed.Moreover,the differences and connections between the two proposed observer methods are discussed based on the connections of actual states,observer states,and the error states.The feedback matrix and observer matrix are obtained by solving linear matrix inequalities.For the nonnegative edge consensus problems with discrete-time positive dynamics,a stochastic matrix is used to describe the interactions of the edges,and the multi-input and multi-output(MIMO)case is considered to release the constraints on the sign of feedback matrices.First,based on state and output feedback method,the necessary and sufficient conditions of nonnegative edge consensus for undirected and directed nodal networks are given.Furthermore,with the properties of nonnegative matrix,sufficient nonnegative edge consensus conditions related with the second largest eigenvalue of the row stochastic matrix,the vertex number,and the edge number of the nodal networks are presented.Based on the property of nonnegative matrix which is Schur stable,the feedback matrix and observer matrix are obtained by solving linear programming problem which reduces the computational complexity.For the edge consensus problem with undirected nodal networks,where the edges are described by continuous-time/discrete-time positive systems with sector input nonlinearities,an observer-based method that can maintain the nonnegative property of the edge states is presented.A rigorous theoretical analysis which considers both the input nonlinearities and the positivity constraints is introduced to achieve a global consensus result.Furthermore,with a modified Lyapunov candidate function and the properties of Metzler matrix,sufficient conditions for continuous-time nonnegative edge consensus are presented with edge system matrices,the edge number,and vertex number of the nodal network.Iterative linear matrix inequalities(LMIs)/LMI algorithm is applied to solve the simultaneous/single stabilization problem to obtain the feedback matrix and the observer matrix.For the observer-based edge consensus problem with directed nodal networks and sector input nonlinearities,the global nonnegative consensus results for continuous-time and discrete-time edge systems are proposed by stabilizing multiple quadratic matrix inequalities with nonnegative constraints.Furthermore,the results are extended to the nonnegative edge consensus with saturation-type sector input nonlinearities,the global and local consensus conditions are discussed.For the continuous-time case,if the nodal network is strongly connected and balanced,the nonnegative consensus conditions are simplified to be solved by stabilizing one quadratic matrix inequality with nonnegative constraints.A multi-step iterative linear matrix inequality algorithm is adopted to obtain the feedback matrix and the observer matrix.