Study Of Sign Pattern Matrix And Consensus Of Multi-agent Systems

In this paper, we study some theories of sign pattern matrix, and consensus problemsof multi-agent system. The study of sign pattern matrices has become somewhat synony-mous with qualitative matrix analysis. We consider some problems of sign pattern, such asallowing diagonalizability. Consensus of multi-agent system is included in research fieldof control theory. We consider improving multi-agent system convergence and collectivecoordinations of multi-agent systems guided by multiple leaders, and so forth. The studyof consensus of multi-agent system needs some theories about matrix and graph. Forthe results obtained about consensus in this paper, we provide some numerical examples.Numerical simulations verify the effectiveness of our theoretical results.We consider some properties of normal sign pattern, and obtain that 2 and 3 ordernormal sign patterns allow diagonalizability. The sufficient and necessary condition judg-ing if a sign pattern allow diagonalizability is still open. In this paper, we obtain somesufficient conditions that sign pattern allowing diagonalizability, and also provide a nec-essary condition. we conjecture that the necessary condition is also a sufficient condition.This conjecture holds depending on the following conjecture: there exists a positive diag-onal matrix such that the production of a nonsingular matrix with this diagonal matrix hask different eigenvalues. We also consider indirect methods to deal with diagonalizableproblem. Moreover, we make a discussion about allowing simultaneously diagonalizabil-ity of two sign patterns. Finally, we discuss the power problem of complex sign pattern.In consensus protocols of multi-agent system, we can often find state differencesbetween a agent and its neighboring agents. However, we find that using the differencesbetween r times state of the agent and states of its neighboring agents can achieve quicklyconvergence by choosing a appropriate parameter r. Therefore, in this paper a new con-sensus protocol taking the differences between r times state of the agent and states of itsneighboring agents is proposed. Based on this idea, two convergence protocols of single-integrator kinematics are considered, and two consensus protocols of double-integratordynamics are also considered. Moreover, some sufficient conditions that these protocolscan converge are obtained. Our consensus protocol has three advantages compared withthe classic one, and our consensus protocol can include the classic one as a special case. In the movement of agents, generally, control input ui(t) of the ith agent plus the termof difference xi(t) - xi(t -Ï„) can achieve quicker consensus. In this paper, we presentsingle-integrator kinematics and double-order-integrator consensus protocols which arebased on this idea. In our results, we consider the case that topology is digraph andLaplacian matrix is asymmetrical. The numerical results show that these protocols havehigher convergence speed.The sufficient conditions which consensus under leader-following protocol withmemory can reach for both continuous-time and discrete-time single order multi-agentsystem are obtained. The sufficient condition which consensus under leader-followingprotocol with memory can reach for a discrete-time single order multi-agent system isalso obtained, which can make system obtain fastest convergence by choosing a appropri-ate parameter r. Likewise, for a leader-following protocol with memory of second ordercontinuous-time multi-agent systems, the sufficient conditions which consensus can reachare obtained.For second-order and high-order dynamic multiagent systems with multiple leaders,the coordination schemes that all the follower-agents -ock to the polytope region formedby multiple leaders are considered. Necessary and sufficient conditions which the fol-lower agents can enter the polytope region by the leaders are obtained.