Delay-dependent distributed event-triggered tracking control for multi-agent systems with input time delay

Abstract

This paper addresses delay-dependent distributed event-triggered tracking control for general linear multi-agent systems. First, both fixed and time-varying input delays are taken into consideration of tracking consensus protocol design due to the network transmission delay among agents, respectively. Correspondingly, a new delay-dependent triggering mechanism is proposed, by which the protocols of agent are only updated at the violation of the pre-defined triggering mechanism. One of its advantages is that the triggering instant can not be affected by its neighbours’ state of last triggering instant, which can reduce effectively the frequency of control updates. Then, the co-design method of the control protocol gains and the triggering parameters are developed and delay-dependent event-triggered tracking consensus criteria are obtained by Lyapunov stability theory, respectively. Moreover, the Zeno-behavior of event-triggered mechanisms is excluded effectively by showing the existence of the strictly positive consecutive triggering interval. Last, two comparison examples are provided to demonstrate the effectiveness of the developed method.

Introduction

Over the past decade, the multi-agent systems (MASs) have aroused much more attention in various areas, such as unmanned aerial vehicles [1], mobile robotics [2], complex network control [3] and microgrids optimization problems [4], and some references therein. The consensus is one of the essential issues in studying MASs and has got extensive research, such as [5], [6], [7], [8], which is similar to the stability of closed-loop control systems [9], [10], [11]. A key question of consensus in the MASs is how to exploit a suitable information exchange algorithm among adjacent agents to make the system stability. In other words, all agents can achieve a collective goal or the same trajectory with the neighboring information only.

Continuous or smaller fixed periodic sampling communication between neighboring agents was usually adopted in the control strategies developed for the MASs to guarantee accurate control performance [1], [12]. However, these continuous or periodic time sampling approaches, usually independent of the system state, may lead to “unnecessary” waste of communication resources. The event-based approach, as an alternative one, initially proposed by Seto et al. [13] for real-time control system, can effectively deal with such problem and has been arousing a great of interest, see Dimarogonas et al. [14], [15], [16], [17], [18]. It reduces effectively the controller updates by permitting updating control information only when the defined measurement error exceeds the specified threshold. There are decentralized and distributed event-based control strategies for the MASs. In [14], [19], decentralized event-based control strategies for first-order dynamics systems and second-order dynamics systems were studied, respectively, in which the event-triggered mechanism depended on the system states. Since the triggering information of its neighboring agents is employed to the agent triggered, it leads to the smaller triggering interval and may appear Zeno behavior. The distributed event-triggered mechanism, which is independent of the states of its neighbor triggering, is proposed first in [20]. It can reduce effectively the times of control updates and the communication bandwidth and be easily applied in practice. In [21], the combinational measurements method was applied into event-triggered control strategy for single-integrator multi-agent systems. In [22], an event-triggered control scheme with distributed, asynchronous and independent features was proposed to the consensus of general linear MASs. In [23], a novel set-membership leader-following consensus was first proposed for the MASs with an unknown-bounded process and measurement noise and the distributed event-triggered mechanism was with a time-varying threshold parameter.

Data transmission delay and dropout in networked control systems are inevitable and affect seriously the control performance, especially in multi-agent systems. Moreover, the less information is exchanged between the agent and its neighboring agents, the more sensitive of consensus performance is to delay. Recently, the input delay control based on event-triggered mechanism has become an attractive research area in MASs. In [15], the network transmission delay was considered into the controller design and co-design approach of the protocol gain and the event-triggered parameters was proposed. Based on the event-triggered and fixed input delay, the single-integrator dynamics and second-integrator dynamics were investigated in [24], [25], respectively. Moreover, for a general linear MASs, the decentralized and distributed control strategies with fixed delay input were proposed in [26], [27], respectively. In [28], the authors studied the robust consensus problem for discrete-time systems with unknown constant time delay input. It is pointed out that all the aforementioned literatures are fixed input delay, which may be very restrictive for the real-world engineering systems. However, the time-varying input delay, which is more general for the networked control system, see Gao et al. [29], [30], has not been paid much attention for MASs. Wang et al. [30] considered the networked synchronization control problem of coupled dynamic networks with time-varying delay. For single-integrator continuous-time MASs, Wang et al. [31] proposed the state dependent and state independent event-triggered consensus control strategy with time-varying input delay, respectively. In [32], the authors mainly discussed based on distributed event-triggered the mean-square consensus for the single-integrator discrete-time MASs with time-varying input delay. As far as we know, the researches on the distributed event-triggered tracking control of MASs with fixed and/or time-varying input delay have not been adequate, let alone on the general linear MASs, which motivates the present investigation.

In this paper, a new delay-dependent distributed event-triggered tracking control was investigated for general linear multi-agent systems. There are mainly three challenges as follows: (1) How to design a suitable controller for general linear multi-agent systems to improve the system consensus performance affected by the networked communication delay among agents; (2) How to construct an appropriate event-triggered mechanism to reduce the frequency of control updates and avoid the Zeno behavior; (3) How to develop an effective method to obtain the control feedback gain and the triggering parameters to improve the tracking performance of MASs?

Motivated by the above-mentioned challenges, in this paper, our contributions are summarized as follows: (i) Both fixed and time-varying input delay control strategies are considered for general linear multi-agent systems, respectively, which can alleviate the effect of communication delay to the MASs and ensure the performance of tracking consensus; (ii) A new delay-dependent distributed event-triggered mechanism is constructed to reduce the frequency of control updates and Zeno behavior is excluded strictly; (iii) The feedback gains of fixed/time-varying input delay controllers and triggered parameters for general linear MASs are co-designed respectively, which ensure effectively the consensus performance and event-triggered updates of MASs.

The rest of this paper is organized as follows. Section 2 presents some definitions and problem descriptions; The main results are developed in Section 3. Two comparison examples are provided in Section 4 and conclusions are drawn in Section 5.

Notations: R^m × n denotes m × n dimensional matrices and I_n represents the identity matrix of n. P > 0 indicates that P is a positive definite matrix. λ_M(P) and λ_m(P) are defined as the maximum value of eigenvalue and the minimum value of eigenvalue for the matrix P, respectively. The superscript T represents the transpose of the matrix. The notation ⊗ is the Kronecker product. ‖ · ‖ is the Euclidean norm for vectors. The notation “⇔” means if and only if.