High-order Linear Multi-agent Systems Time-delayed Consensus Theory And Its Applications To Multi-UAVs Cooperative Control

As a newly proposed theory which only has three years research history, consensus problems of high-order linear multi-agent systems with time delays play a pivotal role in potential applications such as multi-UAVs cooperative control under communication delays. However, some problems in high-order linear consensus with time delays are still open. In the current paper, some of these open problems are studied. By proposing a new mathematical concept-- Kronecker Basis, these problems are finially solved. Additionally, some corresponding applied research in multi-UAVs cooperative control can be done. Mainly, the contributions and novel features are as follows:(1) Kronecker Basis is proposed as a new mathematical theory, and strict definition and corresponding properties are given. Kronecker Basis can be regarded as a generalized Basis by Kronecker Product. When used in high-order linear consensus problems with time delays, Kronecker Basis has more advantages over traditional Basis in describing the Consensus Subspace and Consensus Complement Subspace. As a result, the complexity of high-order consensus problems is reduced and a relatively good foundation is established on solving those open problems.(2) A less conservative convergence criterion on high-order linear consensus with uniform time delays is given. By using Kronecker Basis, consensus convergence criterion with uniform time delays is converted to stability analysis problem with these time delays. Then with adding Free-Weighting Matrices(FWM) in the derivatives of Lyapunov-Krasovskii functional candidates which are used in analyzing the stability problem, the LMIs(Linear matrix inequality) based convergence criterion is established. The numerical examples and simulation results show that compared with existing literatures, the convergence criterion proposed can give larger allowable upper bound in time delays.(3) The convergence criterion on high-order linear consensus with non-uniform time delays is first proposed. By using Kronecker Basis, consensus convergence criterion with non-uniform time delays is converted to stability analysis problem with these time delays. With optimizing the structures of FWMs which are added into the Lyapunov-Krasovskii functional candidate, the high-order linear consensus convergence criterion with non-uniform time delays is firstly established. For evaluating the conservativeness of this criterion, the consensus criteria on first-order multi-agent systems with non-uniform time delays and high-order linear multi-agent systems with uniform time delays in published papers(including our high-order linear consensus convergence criterion with uniform time delays) are compared to the non-uniform time delays criterion as special cases. The numerical examples and simulation results show that the non-uniform time delays criterion has less conservativeness which is mainly due to the optimized structures of FWMs.(4) The necessary and sufficient conditions for high-order average consensus with time delays are first established which are based on the consensus state trajectories given by Kronecker Basis theory. In the first place, the necessary and sufficient conditions for high-order linear average consensus with uniform/non-uniform time delays under balanced communication topologies are given. Accordingly, the previous studies for first-order average consensus with time delays can be regarded as special cases compared to our results. More importantly, the reason why the classic disagreement vector method cannot be adequate to high-order linear average consensus with time delays any more. This can also explain why the studies for average consensus with time delays still stay in first-order integrator multi-agent systems. Finally, the necessary and sufficient condition for high-order linear average consensus with uniform time delays under imbalanced communication topologies is proposed. This necessary and sufficient condition under imbalanced topology can take that under balanced topology as a special case. As a result, this condition can be used under directed topology for high-order linear average consensus with uniform time delays and the balance assumptions in previous papers can be removed.(5) The less conservative protocol parameters designs for high-order linear consensus with uniform time delays are proposed. By employing Kronecker Basis theory, the consensus protocol parameters designs are converted into designing the corresponding stabilization controllers. With the help of congruent transformations, the LMIs based consensus convergence criterion for high-order multi-agent systems with uniform time delays can be a group of NLMIs(Nonlinear matrix inequality) which can be used in calculating the protocol parameters. However, there exists a great difference between undirected and directed communication topologies when solving the corresponding NLMIs. For undirected communication topology, no constraints are added in the FWMs, therefore, the NLMIs can be solved directly by using Cone Complementarity Linearization(CCL); while, for directed communication topology, the elements of FWMs should be restricted. Consequently, this is equivalent to nonlinear constrains in NLMIs which cannot be solved by using CCL. In order to solve this problem, the structures of FWMs are determined in advance and this technique can make it possible to solve NLMIs by CCL. The numerical examples and simulation results show that compared with the existing literatures, our methods have less conservativeness in parameters designs, and have more robustness when the rate of time delays are fluctuating.(6) By employing the proposed high-order linear multi-agent systems time-delayed conseusus theories, the multi-UAVs cooperative control problems are studied, including formation keeping control, multi-UAVs simultaneous arrival and synchronization of multi-UAVs’ observation data under communication time delays. For formation keeping control, the consensus convergence criterion is employed in linearized UAV model to give the allowable upper bound in time delays. If the upper bound is out of the stable range, the controller parameters can be redesigned by consensus protocol parameters designs. For simultaneous arrival of multi-UAVs, the consensus convergence criterion is employed to give the allowable upper bound in time delays for cooperative variable. In addition, this upper bound is compared to that in simulation results to validate the effectiveness. For synchronization of observation data, the consensus convergence criterion is used to analyze the allowable upper bound in keeping data synchronized. Simulation results validate the effectiveness of our theorem.