Study On The Controllability Of Multi-agent Systems With Time Delay

Cooperative control of the multi-agent systems is an important branch of distributed artificial intelligence,which is widely applied in formation control,parallel computing,and data fusion,etc.Technology of cooperative control is the core of multi-agent systems research and controllability is a fundamental issue of cooperative control.This paper studies the controllability of multi-agent systems with time delay,including relative controllability under fixed and switching topologies and structural controllability under fixed topology,respectively.Main contents of this paper are as follows.Firstly,relative controllability of the multi-agent systems with state delays and fixed communication topology is considered.By constructing neighbor-based communication protocols,the homogeneous multi-agent systems and the heterogeneous ones with state delays are transformed into the single-delay systems and the multi-delay ones,respectively.By using Kronecker product and matrix delayed exponential,exact solution of the single-delay systems with permutable parameter matrices is constructed.Based on the solution,the criteria of Gramian matrix and rank of controllability matrix on relative controllability are established,respectively.For the multi-delay systems without parameter matrices permutation,a matrix sequence is constructed by using the parameter matrices and a Mittag-Leffler type matrix function is obtained by using the foregoing matrix sequence.The systems are solved by using the constructed Mittag-Leffler type matrix function,which results in the Gramian matrix and rank of controllability matrix criteria on relative controllability of the multi-delay systems being established,respectively.Secondly,relative controllability of the multi-agent systems with input delay and switching topologies is considered.With neighbor-based protocol adopted,the original problem is transformed into relative controllability of the switched systems with commensurate delays.The switched systems are decomposed into a series of homogeneous and nonhomogeneous systems,and the iterative relations of the solutions of homogeneous and nonhomogeneous systems are constructed,respectively,combining with the switching rule.To establish the criterion on relative controllability for the switched systems,new nested spaces are constructed by using the iterative relations of solutions of the nonhomogeneous systems.Lastly,structural controllability of the multi-agent systems with input and communication delays under fixed communication topology is studied,respectively.Under relative communication protocol,the multi-agent systems with input delay are transformed into the commensurate-delay systems with parameters dependent on one another.Sufficient and necessary conditions for relative controllability of the delay systems are established by using the state space analysis method,whereafter graph-theoretic interpretation for structural controllability of the delay systems is proposed with the approaches of linear parameterization and graph theory utilized.By using absolute protocol,the multi-agent systems with communication delay are transformed into the delay systems with parameters independent on one another.Solution of the delay systems is obtained by using Bellman’s method and algebraic criterion on relative controllability of the systems is established.On this basis,graph-theoretic interpretation for structural controllability of the delay systems is proposed by using the knowledge of graph theory.Corresponding work is extended to the higher order multi-agent systems with communication delay.This research lays the groundwork for future research on optimal control of delay multi-agent systems.