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In this paper, the target controllability of multiagent systems under directed weighted topology is studied. A graph partition is constructed, in which part of the nodes are divided into different cells, which are selected as leaders. The remaining nodes are divided by maximum equitable partition. By taking the advantage of reachable nodes and the graph partition, we provide a necessary and sufficient condition for the target controllability of a first-order multiagent system. It is shown that the system is target controllable if and only if each cell contains no more than one target node and there are no unreachable target nodes, with $\delta-$reachable nodes belonging to the same cell in the above graph partition. By means of controllability decomposition, a necessary and sufficient condition for the target controllability of the system is given, as well as a target node selection method to ensure the target controllability. In a high-order multiagent system, once the topology, leaders, and target nodes are fixed, the target controllability of the high-order multiagent system is shown to be the same to the first-order one. This paper also considers a general linear system. If there is an independent strongly connected component that contains only target nodes and the general linear system is target controllable, then graph $\mathcal{G}$ is leader-target follower connected.

In this paper, the graph based condition for the controllability of game based control system is presented when the control of regulator is not zero. A control framework which can describe realism well expressed as the game based control system (GBCS), was obtained in 2019, which, unfortunately, is not graph theoretically verifiable, and the regulator control input is assumed to be zero. However, based on a new established notion, strategy matrix, we propose a graph theory condition to judge the controllability of GBCS, instead of using algebraic conditions for complex mathematical calculations. More specifically, to tackle these issues, one needs to study the expression of Nash equilibrium actions when regulators control is not zero first. Based on this expression, the general formula of game controllability matrix is obtained, which provides theoretical support for studying the essential influence of topology on game based control system. The general formula is always affected by the specific matrix strategy matrix, composed of Nash equilibrium actions, and the matrix can not only be obtained by matrix calculation, but also can be directly written through the topology, which is the specific influence of the topology on the GBCS. Finally, we obtain the result of judging the controllability of the system directly according to the topological structure, and put forward the conjecture that there is no limitation of equivalent partition in GBCS. Arguably, this is a surprising conjecture on the equivalent partition of graphs, because only the limitation of equivalent partition in fivenode graphs has been solved so far

In view of the problems of large consumption of communication and computing resources in the control process, this note studies a fundamental property for a class of multi-agent systems under event-triggered strategy: the S-stabilizability of a group of multi-agent systems with general linear dynamics under weakly connected directed topology. The results indicate that the S-stabilizability can be described in some way that the stabilizability region and feedback gain can evaluate the performance of the protocol. Firstly, a new distributed event-triggered protocol is proposed. Under this protocol, a kind of hybrid static and dynamic event-triggered strategy are presented, respectively. In particular, by using Lyapunov stability theory and graph partition tool, it is proved that the proposed event-triggered control strategy can guarantee the closed-loop system achieve S-stabilizability effectively, if at least one vertex in each iSCC cell receives information from the leader, which reflects the ability of distributed control law. Further, we demonstrate that the stabilizability can be realized if the initial system matrix A is Hurwitz. Moreover, it is confirmed that the designed static event-triggered condition is a limit case of dynamic event condition and can guarantee Zeno-free behavior. Finally, the validity of the theoretical results is proved by numerical simulation.

In this paper, classic controllability and structural controllability under two protocols are investigated. For classic controllability, the multiplicity of eigenvalue zero of general Laplacian matrix $L^*$ is shown to be determined by the sum of the numbers of zero circles, identical nodes and opposite pairs, while it is always simple for the Laplacian $L$ with diagonal entries in absolute form. For a fixed structurally balanced topology, the controllable subspace is proved to be invariant even if the antagonistic weights are selected differently under the corresponding protocol with $L$. For a graph expanded from a star graph rooted from a single leader, the dimension of controllable subspace is two under the protocol associated with $L^*$. In addition, the system is structurally controllable under both protocols if and only if the topology without unaccessible nodes is connected. As a reinforcing case of structural controllability, strong structural controllability requires the system to be controllable for any choice of weights. The connection between father nodes and child nodes affects strong structural controllability because it determines the linear relationship of the control information from father nodes. This discovery is a major factor in establishing the sufficient conditions on strong structural controllability for multi-agent systems under both protocols, rather than for complex networks, about latter results are already abundant.

In this paper, the controllability and observability of linear multi-agent systems over matrix-weighted signed networks are analyzed. Firstly, the definition of equitable partition of matrix-weighted signed multi-agent system is given, and the upper bound of controllable subspace and a necessary condition of controllability are obtained by combining the restriction conditions of the coefficient matrix and matrix weight for the case of fixed and switching topologies, respectively.The influence of different selection methods of coefficient matrices on the results is discussed. Secondly, for the case of heterogeneous systems, the upper bound of controllable subspace and the necessary condition of controllability are obtained when the dynamics of individuals in the same cell are the same. Thirdly, sufficient conditions for controllable and uncontrollable union graphs are obtained by taking advantage of the concept of switched systems and equitable partitions, respectively. Finally, necessary condition of observability is obtained in terms of the dual system and the constraints of the coefficient matrix, and the relationship between the observability and the controllability of the matrix-weighted signed multi-agent systems is discussed., Comment: 17 pages, 5 figures