Consensus for heterogeneous networked multi-agent systems with switching topology and time-varying delays
Introduction
Consensus seeking of networked multi-agent systems (NMAS) has been a hot topic in the last decade, due to its various applications like coordinated control [1], formation control [2], convex optimization [3]. Generally, consensus seeking constructs communication and control rules (consensus protocol) to synchronize the agents to a common behavior. From dynamics perspective, studies on consensus of networked multi-agent system fall into two categories: studies on consensus for homogeneous multi-agent system and studies on consensus for heterogeneous multi-agent system. Moreover, studies on consensus of multi-agent systems mainly follow two lines: the study of integrator multi-agent systems, and the study of LTI multi-agent systems, see [4] and references therein.
The studies of consensus for homogeneous networked multi-agent system have been flourish in the past decade. From communication perspective, there are two main issues that affect consensus behavior: topology switching and network delay. Since the birth of consensus protocol designing framework proposed by Olfati-Saber and Murray [5], there have been various consensus related results concerning topology switching [6], network delay [7], [8], complex networks [9] and non-linearities [10]. Moreover, the tools for consensus protocol designing are not limited to matrix theory [5], there have been other approaches like contraction method [6], [7], Lyapunov approach [11], passivity method [9], [12], to name just a few. In particular, predictive approach also proved an effective approach in synchronizing the states of agents [13], [14]. There are also other issue concerning consensus like saturation and disturbance [15]. Meanwhile, many new branch conceptions about consensus were also studied, scaled consensus [16], mean square consensus [17], group consensus [18], for example.
Still, consensus seeking is a challenging task for heterogeneous multi-agent systems. The heterogeneity makes it difficult to synchronize the agents in a distributed manner, moreover, the switching-topology and delayed communication environments complicate the situation. There exists much progress on studies on consensus of heterogeneous multi-agent system. To name a few, [19] investigated the consensus problem for heterogeneous multi-agent system of first- and second-order integrator dynamics; [20] studied the consensus problem for heterogeneous multi-agent systems of high-order integrator dynamics and considered the mismatched uncertainties; for general linear dynamics, [21] proposed a distributed adaptive consensus protocol and considered non-linear uncertainties; for the case when topology switches, [22] proposed a consensus protocol for heterogeneous multi-agent systems of first- and second-order integrator dynamics; for the case with both topology switching and network delay, [23] investigated the discrete-time heterogeneous multi-agent systems composed of first- and second-order integrator agents, and constructed two stationary consensus algorithms respectively. Reviewing the recent literature on heterogeneous LTI multi-agent systems, it remains a difficult task to construct a distributed consensus protocol. In particular, when topology switching and time-varying delays are added to the communication simultaneously, the problem becomes even more difficult and remains unsolved so far.
Up to now, most recent works on consensus are focused on single-stage network control. Single-stage network control is reported to have some un-captured functional details in synchronization [24], [25], yet two-stage/multi-stage network control has the advantage of simplifying the control design process [26]. Reviewing the works on two-stage network control, there are mainly two types of methods to implement two-stage/multiple-stage network control. The first is to use subgraphs. For example, Xu et al. [27] splits the overall graph into subgraphs and makes overall decision hierarchically. However, with topology switching, the connectivity of each of the spitted subgraphs is hard to verify and it is not easy to split the graph into dynamic subgraphs. The other way is to use tracking references. For example, in [28] the tracking references of the agents are generated in one layer and tracked by each agent in the other layer, respectively. Yet with heterogeneousness, topology switching and delay, it is not easy to generate dynamic distributed tracking references for each agent in a single layer without leaders.
This paper investigates consensus problem for LTI networked multi-agent systems with switching topology and time-varying delays. A novel two-stage distributed consensus protocol is proposed to solve the problem, including the update for virtual system matrix and the update for the states of the agents, and different from the two two-stage network control strategies above, no graph splitting or reference generating is not needed, both stages are implemented in a distributed manner. Seminorm based consensus analysis is performed and sufficient conditions for consensus convergence are derived for case with periodic topology switching and time-varying delays. The results allow for unstable mode, time-varying delays, and spanning-tree type topology switching, numerical examples illustrates the effectiveness of the theoretic results.
This paper is organized as follows. The first section gives literature review. Some preliminaries of graph theory and seminorm are reviewed in the second section. Section 3 gives the main results. A numerical example and a practical example are presented in the next section. Section 5 concludes the paper.
Section snippets
Preliminaries
To model a multi-agent system, a directed graph is introduced with a node set indexed by an edge set and a non-negative weighted matrix . It means that there exists communication from agent i to agent j if aji > 0 and no communication otherwise. Node j is said to be a neighbor of node i if aij > 0, and the set of agent i’s neighbors is denoted by Ni. is said to be connected if there exists an sequence of ordered edges of form where
Problem Statement and convergence basics
Consider an NMAS consisting of n agents, where the dynamics of agent i is given as follows where are the state, the output, and control input of agent i at t, respectively; Ai, Bi, Ci are matrices with proper dimensions. Let .
This paper considers the state synchronization problem of system (1), and seeks ui(t) such that General communication is considered in this paper. First, each
Simulation
This section provides a numerical example for validation of the proposed protocol. Consider a multi-agent system of four agents with parameters given as follows
The initial configuration of Ti is with ‖Ti‖ ≤ 1.004, given by
The graph is set to have a joint spanning tree, given by where f
Conclusions
Consensus problem lies in the center of distributed control and optimization problem and is difficult to solve because of delays and switching topology within. For heterogeneous LTI multi-agent systems with variable delays and switching topology, pseudo predictive scheme proved a successful tool to overcome delay; seminorm proved to be an effective method for consensus analysis; the two-stage consensus protocol solved the consensus problem in this case in a distributed way. Our next step will
Acknowledgment
This work was supported in part by the National Natural Science Foundation of China under Grants 61333003 and 61773144.
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