Jianxiang Xi

Formation Control for High-Order Linear Time-Invariant Multiagent Systems With Time Delays

Formation control problems for high-order linear time-invariant multiagent systems with time delays are investigated. First, a general time-varying formation control protocol is proposed. Then, based on consensus approaches, necessary and sufficient conditions for multiagent systems to achieve a given time-varying formation are presented. An explicit expression of the time-varying formation reference function is also given. It is shown that the motion modes of the formation reference can be specified. Furthermore, necessary and sufficient conditions for formation feasibility are proposed. An approach to expand the feasible formation set and an algorithm to design the protocol for multiagent systems to achieve time-varying formations are provided, respectively. Finally, numerical simulations are presented to demonstrate theoretical results.

Output consensus analysis and design for high-order linear swarm systems: Partial stability method

Output consensus analysis and design problems for high-order linear time-invariant swarm systems with directed interaction topologies are investigated. Firstly, as foundations of our approaches, several conclusions about partial stability are given. Then, two subspaces of the output space of swarm systems, namely an output consensus subspace and a complement output consensus subspace, are introduced. Based on output projection onto the two subspaces and partial stability, necessary and/or sufficient conditions for output consensus and limited-control-energy consensus are proposed respectively, an explicit expression of the output consensus function is presented based on the different contributions of initial states of agents and protocols, and an approach independent of the number of agents is shown to determine the gain matrices of output consensus protocols. Finally, a numerical example is given to demonstrate the theoretical results.

Delay-dependent admissible consensualization for singular time-delayed swarm systems ☆

Abstract Admissible consensus analysis and design problems for high-order linear time-invariant singular swarm systems with time delays are investigated. Firstly, by state decomposition, the admissible consensus problem is transformed into admissible problems of multiple singular subsystems with lower dimensions. Then, linear matrix inequality (LMI) criteria for admissible consensualization are presented, which only involve eight LMI constraints independent of the number of agents. Moreover, an explicit expression of the consensus function which is independent of time delays is presented, and the impacts of protocol states and interaction topologies on the consensus function are revealed. Finally, a numerical example is given to illustrate the effectiveness of theoretical results.

Practical consensus for high-order linear time-invariant swarm systems with interaction uncertainties, time-varying delays and external disturbances

Practical consensus problems for general high-order linear time-invariant swarm systems with interaction uncertainties and time-varying external disturbances on directed graphs are investigated in this article. A dynamic consensus protocol with non-uniform time-varying delays is adopted to deal with the practical consensus problem. Using state space decomposition, practical consensus problems of a swarm system are converted into stability problems of a disagreement subsystem. Based on the Lyapunov–Krasovskii functional approach and the linear matrix inequality technique, sufficient conditions for swarm systems to achieve practical consensus are proposed where the time-varying external disturbance can be in L 2 or L ∞. Numerical simulations are presented to demonstrate theoretical results.

Output consensus for high-order linear time-invariant swarm systems

Output consensus analysis and design problems of high-order linear time-invariant swarm systems are investigated. First, an output consensus subspace and a complement output consensus subspace are introduced. By output projection onto the two subspaces and the partial stability theory, a necessary and sufficient condition for output consensus is presented, and an explicit expression of the output consensus function, which is jointly determined by the interaction topology and initial states of all agents, is given. Especially, it is shown that the output consensus function is completely determined by initial states of all agents if the interaction topology is balanced. Then, an approach to determine gain matrices in consensus protocols of swarm systems is proposed, which has less calculation complexity. Finally, a numerical example is shown to demonstrate theoretical results.

Guaranteed-cost consensus for multiagent networks with Lipschitz nonlinear dynamics and switching topologies: Guaranteed-cost consensus for multiagent networks with Lipschitz nonlinear dynamics and switching topologies

Guaranteed-cost consensus for high-order nonlinear multi-agent networks with switching topologies is investigated. By constructing a time-varying nonsingular matrix with a specific structure, the whole dynamics of multi-agent networks is decomposed into the consensus and disagreement parts with nonlinear terms, which is the key challenge to be dealt with. An explicit expression of the consensus dynamics, which contains the nonlinear term, is given and its initial state is determined. Furthermore, by the structure property of the time-varying nonsingular transformation matrix and the Lipschitz condition, the impacts of the nonlinear term on the disagreement dynamics are linearized and the gain matrix of the consensus protocol is determined on the basis of the Riccati equation. Moreover, an approach to minimize the guaranteed cost is given in terms of linear matrix inequalities. Finally, the numerical simulation is shown to demonstrate the effectiveness of theoretical results.

Leader–follower guaranteed-cost consensualization for high-order linear swarm systems with switching topologies

Abstract Leader–follower guaranteed-cost consensus analysis and design problems for high-order linear time-variant swarm systems with switching topologies are investigated, where two types of switching topologies are considered; that is, each interaction topology in the switching set has a spanning tree, and the union of a series of interaction topologies in a certain time interval has a spanning tree, which is referred to a joint spanning tree. Firstly, the leader–follower guaranteed-cost consensus problem is introduced, which means that all the followers in a swarm system can track the state of the leader and some performance index should be satisfied simultaneously; that is, leader–follower consensus is achieved in a suboptimal manner. Secondly, by the state decomposition, a necessary and sufficient condition for leader–follower consensus is proposed, which only transforms leader–follower consensus problems into asymptotic stability ones but cannot be directly used to determine whether or not swarm systems can achieve leader–follower consensus. Furthermore, in terms of linear matrix inequalities, sufficient conditions of leader–follower guaranteed-cost consensualization for swarm systems with a spanning tree and a joint spanning tree are presented respectively. Finally, two numerical examples are given to demonstrate theoretical results.

Swarm stability for high-order linear time-invariant singular multi-agent systems

Swarm-stability and swarm-stabilisation problems for high-order linear time-invariant singular multi-agent systems with directed networks are investigated. First, necessary and sufficient conditions for swarm stability and asymptotic swarm stability are proposed, which are independent of the dimensions of Jordan blocks of the Laplacian matrix of the interaction topology. Then, an approach is given to determine the absolute motion as a whole, and it is shown that the absolute motion is completely determined by initial states if the interaction topology is balanced. Furthermore, an approach is presented to determine gain matrices for asymptotic swarm stabilisation. Moreover, leader-following swarm-stability and swarm-stabilisation problems are investigated. Finally, numerical examples are given to demonstrate theoretical results.

Admissible output consensus control for singular swarm systems

For swarm systems consisting of agents described by high-order linear singular systems, admissible output consensus problems are investigated. By using observability decomposition and eigenvalue decomposition techniques, a sufficient condition for admissible output consensus and a necessary and sufficient condition for admissible limited control energy output consensus are obtained. Furthermore, it is shown that stabilisability of the agents is a sufficient condition for admissible output consensualisability, and an approach to solve the protocol design problem is presented. Finally, a numerical example is provided to demonstrate the effectiveness of protocol design approach.

Guaranteed cost consensus for multi-agent systems with time delays

Abstract Guaranteed cost consensus problems for multi-agent systems with time delays are considered. The idea of guaranteed cost control is introduced into consensus problems for multi-agent systems with time delays, where a guaranteed cost function is proposed to simultaneously consider the consensus regulation performance and the energy consumption. For fixed topologies and switching topologies, some sufficient conditions for guaranteed cost consensus are given respectively by the state space decomposition approach and the Lyapunov method. Moreover, an upper bound of the guaranteed cost function and the consensus value are determined respectively for the two cases. Numerical simulations are presented to demonstrate the effectiveness of theoretical results.