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In this paper, we study the controllability of multi-agent systems by equitable partition and automorphism. For the case that cells are incompletely connected outside but completely connected inside, a necessary condition for controllability is given from the perspective of the rank of connection matrix. For the case of multiple cells being completely connected outside and incompletely connected inside, in terms of the eigenvalues and eigenvectors of L and L π , several sufficient and necessary conditions for controllability are presented. Once the quotient graph is controllable under single input or all nodes in nontrivial cells are leaders, the lower bound of controllable subspace is determined. Finally, we give the gap between the necessary condition and the sufficient condition for controllability from the aspect of equitable partition. One highlight of the results in this paper is that we show sufficient conditions to judge controllability by equitable partition and automorphism, which, for specific cases, provides one method that how to break through the defect that equitable partition can only obtain necessary conditions. [ABSTRACT FROM AUTHOR]

In this article, the effects of different models on the controllability of leader‐follower dynamics system based on Laplacian matrix are studied, and the controllability results applicable to general model are obtained. Our research is based on three different control modes, namely, broadcast control, UI$$ {U}_I $$ control, and non‐broadcast control, which cover all the control modes in the general model. Specifically, we first identify the relation of different UI$$ {U}_I $$ control, that is, if the system is controllable under i1$$ {i}_1 $$ different control signals, then increasing the number of control signals does not change the controllability of the system. Furthermore, it is proved that in a fixed topology, when the number of control inputs is increased, the controllability of the system is not correlated with the controllability of the original system, which appears counter‐intuitive. But the spectral radius of the Gramian matrix of the controllable system is increased. In addition, we analyze the condition number of Gramian matrix. Based on the above conclusions, we put forward the concept of hierarchical equivalent partition (HEP), and prove that under the general model, the existence of HEP makes the system uncontrollable. Finally, we expect to find a way to make the system controllable under the general model. Therefore, we obtain sufficient conditions for the system to be controllable. As a by‐product, we find that there is no completely uncontrollable graph when the general model with broadcast control is removed. [ABSTRACT FROM AUTHOR]

In this article, by analyzing the eigenvalues and eigenvectors of Laplacian $L$ , we investigate the controllability of multiagent systems under equitable partitions. Two classes of nontrivial cells are defined according to the different numbers of links between them, which are completely connected nontrivial cells (CCNCs) and incompletely connected nontrivial cells. For the system with CCNCs, a necessary condition for controllability is found to be choosing leaders from each nontrivial cell, the number of which should be one less than the cardinality of the cell. It is shown that the controllability is affected by three factors: 1) the number of the links between nontrivial cells; 2) the rank of the connection matrix; and 3) the odevity of the capacity of the nontrivial cells. In the case of nontrivial cells under the equitable partition, there are automorphisms of interconnection graph $\mathcal {G}$ , which induce the eigenvectors of $L$ with zero entries. For the system with automorphisms, by taking advantage of the property of eigenvectors associated with $L$ , we propose several graphical necessary conditions for controllability. In addition, by the PBH rank criterion, the controllable subspaces of the system with different classes of nontrivial cells are compared. Finally, a necessary and sufficient condition for controllability under minimum inputs is given. [ABSTRACT FROM AUTHOR]

This paper studies the controllability of multiagent systems based on path and cycle graphs. For paths or cycles, sufficient and necessary conditions are presented for determining the locations of leaders under which the controllability can be realized. Specifically, the controllability of a path is shown to be determined by a set generated only from its odd factors, and the controllability of a cycle is determined by whether the distance between 2 leaders belongs to a set generated from its even (odd) factors when the number of its nodes is even (odd). For both graphs, the dimension of the controllable subspace is also derived. Moreover, the technique used in the derivation of the above results is further used to get sufficient and necessary conditions for several different types of graphs generated from path and cycle topologies. These types of graphs can be regarded as typical topologies in the study of multiagent controllability, and accordingly the obtained results have meaningful enlightenment for the research in this field. [ABSTRACT FROM AUTHOR]

In this paper, the controllability of multi-agent systems is studied under leader-follower framework, where the interconnection between agents is directed. The concept of leader-follower connectedness is first introduced for directed graph to check controllability, and some graph-theoretic conditions are derived for constructively designed topologies. Then, distance partition and almost equitable partition are employed to quantitatively study the controllable subspaces. Moreover, the relationship between the state invariance of multi-agent systems and the existence of almost equitable partition is discussed. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Recently, graphical characterization of multiagent controllability has been studied extensively. A major effort in the study is to determine controllability directly from topology structures of communication graphs. In this paper, we proposed the concept of controllability destructive nodes, which indicates that the difficulty in graphical characterization turns out to be the identification of topology structures of controllability destructive nodes. It is shown that each kind of double and triple controllability destructive nodes happens to have a uniform topology structure which can be defined similarly. The definition, however, is verified not to be applicable to the topology structures of quadruple controllability destructive (QCD) nodes. Even so, a design method is proposed to uncover topology structures of QCD nodes for graphs with any size, and a complete graphical characterization is presented for the graphs consisting of five vertices. One advantage of the established complete graphical characterization is that the controllability of any graph with any selection of leaders can be determined directly from the identified/defined destructive topology structures. The results generate several necessary and sufficient graphical conditions for controllability. A key step of arriving at these results is the discovery of a relationship between the topology structure of the controllability destructive nodes and a corresponding eigenvector of the Laplacian matrix. [ABSTRACT FROM PUBLISHER]

This paper considers the controllability problem for both continuous- and discrete-time linear heterogeneous multi-agent systems with directed and weighted communication topology. First, two kinds of neighbour-based control protocols based on the distributed protocol of first-order and second-order multi-agent systems are proposed, under which it is shown that a heterogeneous multi-agent system is controllable if the underlying communication topology is controllable. Then, under special leader selection, the result shows that the controllability of a heterogeneous multi-agent system is solely decided by its communication topology graph. Furthermore, some necessary and/or sufficient conditions are derived for controllability of communication topology from algebraic and graphical perspectives. Finally, simulation examples are presented to demonstrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the controllability of first-order and second-order discrete-time multi-agent systems with directed topology and input delay. The problem is studied in the leader–follower framework where a number of agents are selected to be leaders and serve as control inputs to all other agents. Sufficient and necessary conditions are derived for the controllability of first-order discrete-time multi-agent systems. With sampling period and feedback gain satisfying certain conditions, it is proved under three different protocols that the controllability of second-order discrete-time multi-agent systems is equivalent to that of a pair of submatrices of Laplacian matrix. In addition, the controllability of both first-order and second-order discrete-time multi-agent systems with input delay is shown, through some transformations, to be equivalent to that of the transformed non-delayed systems. Finally, some simulation examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: The paper studies decentralized stabilizability for multi-agent systems with general linear dynamics. The stabilizability problem is formulated in a way that the protocol performance can be evaluated by means of the stabilizability region and the feedback gain. For fixed topology, it is proved that the system is stabilizable if and only if external control inputs are exerted on some indicated agents. The result is further shown to be a prerequisite for subsequent design of the corresponding decentralized external self-feedback control, which is also necessary and sufficient. Based on this, two methods are presented to find the agents under which stabilizability can be reached, and the region of stabilizability is given to evaluate the protocol performance. For switching interaction topology, it is shown that the system is stabilizable even if each of its subsystems is not. Finally, the results are employed to cope with the decentralized set-point formation control problem, for which some necessary and/or sufficient conditions are developed. Numerical simulations are presented to demonstrate the effectiveness of the proposed results. [Copyright &y& Elsevier]

Abstract: By controllability of multi-agent systems, we mean the property that all agents can be driven to arbitrary configurations only through regulating a few leaders externally, while the rest of the agents are subject to the nearest-neighbor-law. It is known that the controllability of multi-agent systems depends not only on the communication topologies among agents but also on the selection of leaders. The main objective of this paper aims to characterize the virtue that leaders should have from the perspective of algebraic and graphical conditions. We begin with a heuristic observation for tree graph. To gain more insights, necessary and sufficient conditions are then proposed in terms of downer branch and subgraphs, respectively. In particular, it is shown for path topologies that controllability completely depends on the leaders’ location. The obtained results may help us to pave the way towards a new method of formation and reconfiguration control in swarming robots. [Copyright &y& Elsevier]

Abstract: In this paper, the formation control problem of the network of multiple agents is studied in terms of controllability, where the network is of the leader–follower structure with some agents taking leaders role and others being followers interconnected via the neighbor-based rule. It is shown that the controllability of a multi-agent system can be uniquely determined by the topology structure of interconnection graph, for which the investigation comes down to that for a multi-agent system with the interconnection graph being connected. Based on these observations, two kinds of uncontrollable interconnection topologies are characterized, and a necessary and sufficient eigenvector-based condition is presented. Our studies also touch upon the selection of leaders. [Copyright &y& Elsevier]

Abstract: The paper presents a unified perspective on geometric and algebraic criteria for reachability and controllability of controlled switched linear discrete-time systems. Direct connections between geometric and algebraic criteria are established as well as that between the subspace based controllability/reachability algorithm and Kalman-type algebraic rank criteria. Also the existing geometric criteria is simplified and new algebraic conditions on controllability and reachability are given. [Copyright &y& Elsevier]

Abstract: This paper studies the reachability of discrete-time switched linear systems. A new constructive approach to designing switching sequences is developed so that not only the reachable subspace is realized but also the required number of switchings is reduced significantly. Detailed comparison between the obtained result and that reported in the open literature is also made, which verifies the advantage of the proposed approach. [Copyright &y& Elsevier]

Abstract: In this paper, we study the problem of designing switching sequences for controllability of switched linear systems. Each controllable state set of designed switching sequences coincides with the controllable subspace. Both aperiodic and periodic switching sequences are considered. For the aperiodic case, a new approach is proposed to construct switching sequences, and the number of switchings involved in each designed switching sequence is shown to be upper bounded by Here d is the dimension of the controllable subspace, , where are subsystems. For the periodic case, we show that the controllable subspace can be realized within d switching periods. [Copyright &y& Elsevier]