Event-triggered consensus control for second-order multi-agent system subject to saturation and time delay

Abstract

The event-triggered consensus control for second-order multi-agent systems subject to actuator saturation and input time delay, is investigated in this paper. Based on the designed triggering function, a distributed event-triggered control strategy is presented to drive the system to achieve consensus. Communication energy can be saved as the agents send their state information only at infrequent event instants, the continuous communication among agents is not necessary. Lyapunov-Krasovskii functional is used together with linear matrix inequality technique to analyze the stability of the closed-loop error system. The results show that agents achieve exponentially consensus under the proposed controller. Furthermore, the bounds of solution are obtained by establishing the differential equation associated with the first delay interval. The initial domain is estimated by optimizing the linear matrix inequalities. Finally, simulation examples are presented to illustrate the effectiveness of the proposed controller.

Introduction

Recently, the consensus problem of multi-agent systems (MASs) has been studied due to its wide applications in many fields, such as the sensor/actuator network formation, multi-robot cooperation, unmanned aerial vehicles (UAVs), and so on [1], [2], [3], [4], [5], [6]. The consensus means each agent updates its controller associated with its neighbors’ states and its own state such that the states or outputs are achieved the desired value by using interaction among agents.

Up to date, there are a lot of results available on the consensus problem of MASs from different perspectives [7], [8], [9], [10], [11], [12], [13], in which the event-triggered consensus control is one of the most concerned issues. The so-called event-triggered consensus control is that the controller update only happens when the measurement error increases to a certain threshold rather than execute task periodically and state-independent. Compared with traditional time-based control, event-triggered consensus control has the advantages of reducing update numbers and transmission frequencies in communication network. Since event-triggered control is aperiodic, agents need to communicate with their neighbors when the predesigned triggering mechanism is satisfied. All of these stimulate considerable research efforts on event-triggered control protocols.

However, since the controller drives the object through the actuator in the actual engineering control system, the actuator is inevitably affected by input saturation due to the limitations of the environment and the physical structure of itself. Therefore, it is essential to study the consensus control subject to input saturation. At present, there are a few research results on event-triggered consensus control for MASs with actuator saturation. Some results have been reported [1], [2], [9], [14], [15]. Input saturation constraint was studied for MASs with single integrator dynamics in [9] and with double integrator dynamics in [10], respectively, where the consensus controllers were designed without adopting the event-triggered strategy. In [15], the authors studied leader-following consensus control for MASs with actuator saturation under switching topologies by using the low gain feedback method, while only semi-global consensus was obtained. Similarly, semi-global consensus for MASs with input saturation was also studied in [16], where a self-triggered control protocol that only uses the agent neighbors’ information to update the controller, was proposed. However, the protocol can only be applied to leader-following consensus for MASs. Different from [15], [16], [17], the global leader-following consensus subject to input saturation was achieved not only under fixed undirected network topologies but also under time varying network topologies. However, in [17], [18], [19], only the linear model with single integrator of MASs was considered, which cannot be used directly in MASs with the dynamics of double integrator. Both semi-global stability and global stability were obtained in [17], [18], [19] under their own triggering conditions, but they did not involve the problem of the attraction domain, which is important to systems with actuator saturation.

With the further investigation of the consensus problem, it is worthy noting that time delay is included in consensus because of its inevitability in actual applications. It is known that the input time delay can deteriorate system performance and lead to system instability from communication channels and external environment. Thus, it is essential to study the event-triggered consensus control for MASs with input time delay. So far, much progress has been made in the investigation of the consensus for MASs with input time delay. In [20], the authors investigated the event-triggered consensus control for second-order MASs with time delay, and the agents achieved consensus exponentially, where an exponential decay term was added to the triggering function, and continuous communication among agents was not necessary. In [21], [22], the authors solved the consensus problem of nonlinear MASs with random disturbance and time-varying delay, but the Zeno-behavior was not taken into account. In [23], the authors only dealt with the problem of solution bounds for MASs with time delay, an additional solution bounds and the first time interval were given by using Lyapunov-Krasovskii functional (LKF) approach. In [24], [25], the leader-following consensus was achieved through applying event-triggered methods to single integrator models of MASs with input time delay. In [26], [27], [28], [29], [30], although consensus problems for MASs with time delay were studied via different methods, the triggering mechanism and saturation system were not investigated. As far as we know, there are few works concerned the attractive domain for second-order MASs with actuator saturation and input time delay, so it is still an open and challenging issue.

In this paper, different from leader-following consensus in [10], [13], we study the event-triggered leaderless consensus control for the second-order MASs. Besides, we propose different consensus protocols which only use the discrete states sampled and sent by neighbors at their event instants. Compared with [6], we propose a linear state-dependent triggering function that can simplify the computation, thus the proposed consensus control strategy can be easily apply into practice. In addition, we investigate the event-triggered consensus control for second-order MASs with input saturation and input time delay. A distributed event-triggered control protocol is proposed. A sufficient condition is given under the designed triggering function. The initial domain is estimated by using LKF method. Different from [8], [12], we consider the weighted directed topological graph with input time-varying delay, LKF techniques combined with linear matrix inequalities (LMIs) are used to analyze the problems. Finally, the simulations are shown to verify the effectiveness of the proposed consensus control protocol. The main contributions of this work are summarized as follows.

(I) The distributed event-triggered control protocol for second-order MASs with nonlinear saturation and input time delay, based on non-periodic sampled data and measurement error, is designed. Compared with the studied in [11], [12], [13], a linear state-dependent triggering function is given, which decreases the computational complexity. With this designed protocol and triggering function, exponential stability of closed-loop error system is achieved.

(II) LKF approach is used together with LMI technique to analyze the convergence of the closed-loop error system, which can transform the consensus problem into the stability problem of Lyapunov function by solving LMI. The results show that the consensus can be achieved exponentially under the proposed control protocol.

(III) For the second-order multi-agent systems with nonlinear saturation and input time delay, the initial domain is required with special consideration, which is closely related to the convergence of the system. In this paper, the solution bounds are obtained by solving the differential equations associated with the first time delay interval. The initial domain is estimated by solving the optimization problem of LMIs.

The rest of the paper is deployed as follows. In Section II, some mathematical preliminaries related to graph theory, several definitions and lemmas are presented. System model and triggering function design are shown in Section III. Main results are shown in Section IV. Simulations are provided in Section V to verify the effectiveness of the theoretical analysis. In Section VI, conclusions are summarized and further study works are proposed.