Asynchronous adaptive event-triggered tracking control for multi-agent systems with stochastic actuator faults

Abstract

This paper investigates the problem of consensus tracking control for multi-agent systems with stochastic actuator faults, where the asynchronous adaptive event-triggered scheme (AETS) is proposed to reduce communication burden. The failure phenomenon for each actuator is described by an independent and identically distributed (i.i.d) random process. The communication information is divided two groups according to the scenario whether the followers can receive the information from the leader or not. Two adaptive event-triggered communication schemes are introduced based on the above classified information. Considering the stochastic actuator faults and the designed AETS, a new closed-loop fault-tolerant tracking control system model is established. Then, by employing the Lyapunov stability theory and stochastic analysis technique, a new fault-dependent stability criterion is derived, which ensures that the leader-following tracking error system is stochastically mean-square stable. Based on this criterion, the fault-tolerant consensus controller gains and the triggering parameters are obtained simultaneously by solving a linear matrix inequality (LMI). Moreover, the Zeno behavior of the AETS is excluded. Finally, a simulation example is included to illustrate the effectiveness of theoretical results.

Introduction

In recent years, multi-agent systems (MASs) has received substantive attention due to its potential applications in many fields, such as sensor networks, unmanned ground/air vehicles, spacecraft formation flying, cooperative surveillance, radial DC micro-grid and so on [1], [2], [3], [4], [5], [6]. In cooperative control of MASs, the consensus problem is a fundamental problem, which has been widely studied [7], [8], [9], [10], [11], [12]. Specifically, the consensus tracking control for MASs has been investigated as a paradigm for some specific behaviors by designing some consensus algorithms to ensure all the followers can reach the states of leader as much as possible.

In practical applications of networked systems, controller actuation is only updated at some discrete time instances. However, a wireless medium is often exploited to communicate enormous data packets in networked systems, which may introduce network-induced delays and packet dropouts because of limited bandwidth. On the other hand, the traditional sampling method may often introduce unnecessary updating as well as actuator attrition. In order to save the network resources and reduce unnecessary communication, some novel sampling methods such as aperiodic sampling, asynchronous sampling method, round-Robin scheduling method, stochastic sampling method and the event-triggered sample mechanism have recently received considerable interest in the research of community control theory, and some interesting works have been reported in literature, see, e.g., [13], [14], [15], [16], [17], [18], [19], [20], [21], [22].

More recently, numerous research works concerning the event-triggered consensus or synchronization for MASs have been found [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39]. Distributed event-triggered models were adopted to investigate the consensus problem of MASs in [23], [24], [25], [26], [27], [28]. Based on the centralized event triggering condition, the leader-following consensus was studied in [29], [30]. By predicting the relative inter-node states, an input-based triggering control strategy is concerned for the synchronization of MASs in [31]. The general linear model in [32], [33], [34], [35] and nonlinear model in [27], [36] were considered to investigate the event-triggered synchronization problems, respectively. The event-triggered leader-following consensus for the second-order MASs was explored in [37]. In addition, some results on observer-based event-triggered consensus for MASs were also developed in [38], [39].

The above mentioned references provided some fundamental works for event-triggered tracking control problems of MASs. However, there still exist some further improvement margins. More specifically, above all the event-triggered consensus control problems for MASs mainly focused on the distributed and centralized event-triggered scheme (ETS). It is noted that the distributed ETS means that every control channel will need an event trigger, which will consume large costs or other expenditures when plenty of agent nodes exist. Centralized ETS means that the updating signals of all agents should be transferred at the same time. However, this centralized ETS will result heavy network congestion, especially when the wireless network bandwidth is limited. Meanwhile, every agent uses the identical triggered condition decided by the information of all agents. Under such a triggering mechanism, some unnecessary data may be triggered while the necessary one may not. In [29], the authors proposed the clustered-based event triggering method to achieve leader-following consensus. In this study, the authors only considered a special case, where the agents in some cluster will not exchange information with the agents in other clusters, i,e, the topology graph induced by followers is an unconnected graph, so this categorizing method is difficult to expand the case of connected graph. Moreover, the above triggered schemes are almost static, where the triggered threshold is a predetermined constant. Thus, it is hard to adapt to the changes of external environmental and internal condition. In recent years, there are some researches on adaptive event-triggered control system, see [40], [41], and the references therein. However, there are few correlated results about designing the adaptive event-triggered consensus protocols for MASs.

To balance the advantages and disadvantages of distributed ETS and centralized ETS, we introduce asynchronous AETS, where the event thresholds can be online adjusted according to changes of $\parallel (L\otimes I_n)\tilde{e}\left(t\right)\parallel$ and $\parallel (D\otimes I_n)\overline{e}\left(t\right)\parallel$ prosed in paper, respectively. The proposed AETS can achieve various control performances under different circumstances. On the other hand, the updating signals from the two triggers are asynchronous, which are hardly amalgamated together at the same time. Thus, the reduction of geometric dimension for the signal being transmitted may largely avoid the network congestion, which will improve the communication performances under network environment. This is our fist motivation to increase network transmission efficiency by improving the traditional ETS.

It deserves to be noted that, with the increasing scale and complexity of MASs, disturbances and faults are unavoidable. The occurring actuator faults will degrade system performances or even make the system paralysis. For the demand of safety and reliability, the study of actuator faults is getting people to pay attention. But the relevant researches on fault-tolerant control has not been extensively explored in the field of multi-agent networks, although there are some papers on considering actuator faults for MASs in the past years [42], [43], [44], [45], [46], [47], [48], [49]. For example, in [42], [46], fault-tolerant tracking control for nonlinear MASs was investigated. In [43], [44], [47], some results of adaptive consensus control were obtained for MASs with actuator faults. In [42], [48], adaptive fault-tolerant tracking control of MASs was discussed, but the actuator faults were only assumed to occur constantly. To the best of our knowledge, few works has considered the stochastic fault case. The fault-tolerant tracking control control for MASs with stochastic actuator faults is still an open issue, which is the second motive of this paper.

As an important control method, feedback control for various systems has been extensively studied [50], [51], [52], [53]. Based on feedback control design method, asynchronous AETS and stochastic actuator faults are considered in a comprehensive way to investigate fault-tolerant tracking control for MASs in this paper. In comparison with the recent works on event-triggered leader-following tracking control for MASs, the main contributions of our method are introduced as:

(1) Compared to the static event-triggered scheme, the proposed AETS can be online adjusted triggering threshold parameters for monitoring system state to achieve control performances under different circumstances. In addition, it is showed that Zeno behavior of AETS is excluded in the absence of actuator faults.

(2) The asynchronous AETS is designed based on two types of information, which contributes more to transmission efficiency than centralized ETS, and lower to cost of trigger than distributed ETS.

(3) Asynchronous AETS and actuator faults are considered in a unified framework to construct a novel tracking control protocol. Using Lyapunov stability theory, the stabilization criterion is derived such that the leader-following tracking error systems are stochastically mean-square stable. The explicit expression of feedback gain and the weighting matrices of AETS are also co-designed, which is one main difference between the present works and existing paper such as [30].

The rest of paper is organized as follows. Section 2 presents the formation of the event-triggered MASs with actuator faults. In Section 3, some sufficient conditions for the stochastic stability of tracking error systems are established in terms of LMI, and a controller design method is provided. A simulation example is employed to demonstrate the effectiveness and applicability in Section 4, and conclusions are drawn in Section 5.

Notation: Rⁿ denotes the n-dimensional Eculidean space, and the superscript “T” stands for matrix transposition; I is the identity matrix of appropriate dimension; $\text{diag}(A_1,A_2,\dots ,A_n)$ represents a block-diagonal matrix with matrices A_i on its diagonal. The notation X < 0 means that the matrix X is real symmetric negative definite. For a matrix B and two symmetric matrices A and C, symmetric matrices of the form $\left[\begin{array}{cc}A& B^T\\ B& C\end{array}\right]$ is written as $\left[\begin{array}{cc}A& *\\ B& C\end{array}\right]$; A⊗B denotes the Kronecker product of matrices A and B. Besides, ‖ · ‖ refers to two-norm for vectors or the induced two-norm for matrices, $\mathbb{E}\{·\}$ stands for the mathematic expectation.