Research On Consensus Of The Linear Multi-agent Systems And Its Global Optimality

In the past two decades,cooperative control for multi-agent systems has re-ceived compelling attention from many scientific communities due to its broad ap-plications in many areas.Complicated tasks can be accomplished by a group of agents with simple structure in a distributed way.Compared with the conventional centralized and decentralized control schemes,it is more efficient,simple and af-fordable.One of the most basic problems of the multi-agent systems is consensus problem which distributed protocols are designed to drive all agents reach a common agreement on certain quantities of interest.The consensus is widely used includ-ing unmanned aerial vehicles(UAVs),distributed sensor networks,etc.Thus,the study on consensus issue is of great significance in both theory and applications.Since there will almost be a discrepancy between the actual plant and the nominal plant used in design and for the good robustness provided by the optimal controller,research on optimal control problems is of great significance.The problem of design-ing the optimal coordination control which not only guarantees the consensus of the agents,but also minimizes some performance indexes is viewed as an open research problem.This dissertations makes the further research on designing distributed con-sensus protocols which satisfy two design requirements for identical continuous-time linear multi-agent systems on fixed,undirected graphs:meeting the global opti-mality and guaranteeing a prescribed convergence speed,using the inverse optimal approach to design distributed consensus protocols that guarantee consensus and global optimality with respect to some quadratic performance indexes for identi-cal continuous-time linear systems on a directed graph,developing an LQR(linear quadratic regulator)optimal design method to solve the consensus problem for i-dentical discrete-time linear systems on a directed graph and showing the sufficient conditions and necessary conditions of the global optimality.The main research of the dissertation can be briefly described as follows:1.Distributed consensus protocols are designed which satisfy two design require-ments for identical general linear multi-agent systems on fixed,undirected graphs:meeting the global optimality and guaranteeing a prescribed conver-gence speed.By using inverse optimal approaches,the optimal partial stabi-lization is developed for linear systems and the globally optimal distributed consensus problem for leader following and leaderless problems are solved.To obtain prescribed convergence speed of the multi-agent system,novel globally optimal distributed consensus design procedures are proposed.First,com-bining with the regional pole assignment,the optimal control can be found by solving a strict linear matrix inequality(LMI)problem.Then,a modified linear quadratic regulator(MLQR)design method is developed.2.For general identical linear continuous-time cooperative systems,we propose two indexes to evaluate the consensus performance:convergence rate and damping rate.The convergence rate is used to evaluate the convergence speed of the agents,and the damping rate is used to evaluate the oscillating behav-iors of the agents.The graph topology is assumed to be fixed and directed.Then a novel distributed protocol design method is proposed,the resulting multi-agent systems can achieve specified convergence rate and damping rate asymptotically and the resulting protocols have the whole right half complex plane as its asymptotical consensus region.3.The inverse optimal approach is employed to design distributed consensus pro-tocols that guarantee consensus and global optimality with respect to some quadratic performance indexes for identical linear systems on a directed graph.The inverse optimal theory is developed by introducing the notion of partial stability.As a result,the necessary and sufficient conditions for inverse opti-mality are proposed.By means of the developed inverse optimal theory,the necessary and sufficient conditions are established for globally optimal coop-erative control problems on directed graphs.Basic optimal cooperative design procedures are given based on asymptotic properties of the resulting optimal distributed consensus protocols,and the multi-agent systems can reach desired consensus performance(convergence rate and damping rate)asymptotically.4.A novel LQR-based optimal distributed cooperative design method is devel-oped for synchronization control of general linear discrete-time multi-agent systems on fixed,directed graphs.Sufficient conditions are derived for syn-chronization based on the relation of the graph eigenvalues to a bounded cir-cular region in the complex plane which couples the agent dynamics and the Riccati solution.The synchronizing speed issue is also considered and it turns out that the synchronizing region reduces as the synchronizing speed becomes faster.To obtain more desirable synchronizing capacity,weighting matrices are selected by sufficiently utilizing the guaranteed gain margin of the optimal regulators.In contrast of the existing work,the synchronizing error of all fol-lowers to the leader converges to zero ultimately,rather than with bounded residual errors.5.Distributed protocols are designed for linear discrete-time multi-agent systems on fixed directed graphs which not only guarantee consensus but also mini-mize some global quadratic performance indexes.Both leaderless consensus and leader-following consensus problems are considered.Partial stability the-ory and inverse optimal control approach are employed to construct sufficient conditions on global optimality in the sense of LQR and derive restrictions on the communication topology.It turns out that if the graph matrix is simple and all its eigenvalues lie inside of a specified interval,then the global opti-mality could be achieved by a linear LQR-based distributed protocol.On the other hand,the global optimality achieved by a linear distributed protocol restricts the graph matrix necessarily to be simple.Finally,concluding remarks are given.Some unsolved problems and develop-ment direction for the multi-agent systems are given.Furthermore,the prospects of the further study are given.