Switching controllability of discrete-time multi-agent systems with multiple leaders and time-delays

Abstract

This paper investigates the switching controllability of discrete-time multi-agent systems with multiple leaders and time-delays on undirected networks, in which both a single time-delay and multiple time-delays are addressed. It is established that the switching controllability of discrete-time multi-agent systems is only determined by the information from the leaders to the followers, and the discrete-time multi-agent systems are controllable even if each of its subsystems is uncontrollable. For the special case of fixed topology, an equivalent augmented system without time-delays is proposed to analyze the controllability of multi-agent systems via PBH rank test. Some numerical examples are provided to demonstrate the effectiveness of the theoretical results.

Introduction

In recent years, distributed coordinated control of multi-agent systems has attracted much attention among researchers from diverse fields [1], [2], which is partly due to the advances in communication and computation, as well as the broad applications of multi-agent systems in many areas including cooperative control of unmanned aircrafts, robots, and satellite clusters [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. Studies in this field have been greatly inspired by the ubiquitous cooperative behavior in real world, such as bird flocks, fish schools, ant swarms and bacteria colonies. In the context of multi-agent networks, however, the issue of controllability presents new features and difficulties [13], [14], [15], [16]. In general, the controllability of a multi-agent network means that the multi-agent system can be steered from any initial state to arbitrary desired state by certain regulations. From the perspective of the design of controller, the systems can obtain any predetermined property when the systems are controllable.

In [17], the issue of controllability of a group of autonomous agents interconnected through nearest neighbor rules was investigated. Necessary and sufficient conditions are obtained based on time-invariantly nearest neighbor topology without time-delays. In [18], the controllability of continuous-time system with a leader was considered. In [19], the controllability of a leader–follower discrete-time dynamic network with switching topology was discussed. An early result on the controllability of discrete-time system with time-delay was given in [20]. Controllability of a leader–follower dynamic network with interaction time-delays was shown in [21]. More discussions about the controllability of multi-agent systems with time-delays are given in [22], [23], [24], [25]. In [26], [27], an operative method was presented to simplify the original systems with time-delays. However, in the above works, the systems with time-delays are composed of one leader or without leaders. Controllability of multi-agent systems was investigated from a graph-theoretic perspective [28]. Based on relaxed equitable partitions, the controllability of multi-agent systems was analyzed in [29]. Formation controllability of multi-agent systems with high-order dynamics was studied in [30].

Based on the unified framework mentioned in [31], [32], we will investigate the switching controllability of discrete-time multi-agent systems with multiple leaders and time-delays. Different from the existing results, both multiple leaders and time-delays are considered in this paper, which presents a feature of practical interest for multi-agent systems and is so complex and difficult to study the controllability of discrete-time multi-agent systems. An agent is called a follower if it updates its state by the control of its neighbor leaders and the information from its neighbor agents, otherwise called a leader. Leaders only receive the information from the outside control input acting as the exogenous control to steer the whole system. The topology among the agents is of great importance in the switching controllability of multi-agent systems. In this paper, we consider an unified structure based on leader-following and undirected structures, in which time-delays occur in the coupling information among the followers. Both a single time-delay and multiple time-delays are considered in the multi-agent system. Some necessary or sufficient conditions have been derived to determine the controllability of the multi-agent systems. The results show that the switching controllability of multi-agent systems does not require the controllability of its subsystems and can only be determined by the information from the leaders to the followers. For a special case, that is, the fixed networked topology, the controllability of multi-agent systems with time-delays can be tested by its eligible subsystems, which is theoretically proved via PBH rank test. The main contributions of this paper are threefold. First, both the multiple leaders and multiple delays of discrete-time multi-agent systems are studied. Second, the PBH test is introduced to justify the controllability of the system. Finally, the controllability of subgroups is considered.

This paper is organized as follows. Section 2 introduces some basic concepts and notations in graph theory. Section 3 presents the formulation of the problems. Section 4 gives some main results via switching topology. Section 5 discusses the system with fixed topology, and some further discussions and relevant results are also given in this section. Section 6 gives some examples and simulations about the controllability of the system. The conclusion is given in Section 7.