Dynamic leader-following consensus for asynchronous sampled-data multi-agent systems under switching topology

Abstract

This paper investigates the leader-following consensus problem for asynchronous sampled-data multi-agent systems with an active leader and under switching topology, in which the asynchrony means that each agent’s update actions are independent of the others’. First, the dynamic leader-following consensus problem for asynchronous sampled-data systems is transformed into the convergence problem of products of infinite general sub-stochastic matrices (PIGSSM), where the general sub-stochastic matrices are matrices with row sum no more than 1 but their elements are not necessarily nonnegative. We develop a method to cope with the corresponding convergence problem by matrix decomposition. In particular, we split the general sub-stochastic matrix into a sub-stochastic matrix which is a nonnegative matrix with row sum no more than 1, and a matrix with negative elements and row sum 0. Then based on a graphical approach and matrix analysis technique, we present a sufficient condition for the achievement of dynamic leader-following consensus in the asynchronous setting. Finally, simulation examples are demonstrated to verify the theoretical results.

Introduction

The cooperative control of multi-agent systems has been experiencing a rapid development in recent years. As an important and fundamental issue in cooperative control, the consensus problem has received increasing attention from various perspectives. Consensus refers to reaching an agreement regarding a certain quantity of interest, and lots of efforts have been made in the design and analysis of various consensus protocols for different system situations [6], [14], [18], [19], just to name a few. In the literature related to the consensus problem, agents are usually considered to be governed by first-order dynamics. Meanwhile, there is a growing interest in consensus algorithms which take the form of second-order dynamics, partly due to its ability to model a broader class of complicated agents. For example, some mobile robot dynamics can be feedback linearized as double integrators. By analyzing eigenvalues of the corresponding system matrix, Yu et al. [28] gave some necessary and sufficient conditions for second-order consensus of multi-agent systems with fixed topology. For a class of second-order continuous-time multi-agent systems with time-delay and jointly-connected topologies, Lin and Jia [11] derived a sufficient condition in terms of linear matrix inequalities (LMIs) for average consensus. In [10], Li et al. discussed the second-order nonlinear consensus in general directed networks, and presented some algebraic criteria to unravel the underlying mechanics for reaching consensus. For second-order multi-agent systems under limited interaction ranges, Ai et al. [1] studied a distributed linear consensus protocol and derived a sufficient condition to achieve consensus by using a Lyapunov functional approach. By employing the sliding mode control method, Qin et al. [16] investigated the consensus tracking problem of second-order nonlinear multi-agent systems with disturbance and actuator fault.

Most of the aforementioned works are concerned with synchronous consensus of second-order multi-agent systems with fixed topology, where all the agents are assumed to share the same clock and process their data synchronously, and the communication topology between the agents is assumed to be unchanging. In practical applications, the asynchronous property in signal transmission is more prominent in multi-agent systems, which means the central synchronizing clock may not be available and each agent’s update actions are independent of the others’. In addition, the communication topology may be dynamically changing due to the unreliability of information channels and the limited communication range of agents, so it is practically preferable that each agent updates its actions according to its own schedule and under time-varying networks. These kinds of realistic scenarios raise the problem of asynchronous consensus with switching topology [7], [15], [20], [21], [22], [23], [24], [25], [29]. For instance, In [7], Gao and Wang studied the asynchronous consensus problem of continuous-time second-order agents with time-varying delays, where a sufficient condition in virtue of the Lyapunov’s direct method was established for asynchronous consensus. Based on nonnegative matrix theory and graph theory, Xiao and Wang studied asynchronous consensus problems for the multi-agent systems with intermittent information transmission and time-varying delays in [23]. In [22], they further addressed the partial state consensus problem of multi-agent systems with second-order dynamics, and proposed an asynchronous consensus protocol for the case with switching topology and time-varying delays. Along this research line, Xiao et al. presented a Lyapunov-based stability result for asynchronous sampled-data multi-agent networks with time-varying delays in [25]. By using tools from nonnegative matrix theory and graph theory, Qin et al. [15] investigated the asynchronous consensus problem of discrete-time second-order multi-agent systems under switching topology. In [21], Shi et al. investigated the asynchronous group consensus problem for discrete-time heterogeneous multi-agent systems under dynamically changing topologies.

For investigating asynchronous consensus problem of multi-agent systems with switching topology, one typical method is constructing common quadratic Lyapunov functions [7], [25]. However, even for the simpler synchronous consensus problem with switching topology, it is highly demanding to construct a common Lyapunov function especially for the case with nonexistence of a common left eigenvector for all system matrices, which was in particular demonstrated in [12] for the discrete-time multi-agent systems. Besides, the derived conditions are usually given in terms of LMIs in the common Lyapunov functions method, which will lead to higher computational complexity for the solvability of LMIs when the scale of network increases. Another typical method for asynchronous consensus problem with switching topology is utilizing nonnegative matrix theory, especially the convergence property of products of infinite stochastic matrices [15], [22], [23], [24], [29]. In detail, by employing specific linear transformation, the asynchronous consensus problem with switching topology is converted into the synchronous consensus problem of the augmented systems, whose convergence analysis mainly relies on the property of products of infinite stochastic matrices. Then apply the classic result of Wolfowitz to derive the conclusion. Unfortunately, the existing Wolfowitz result is build upon a relative strong assumption that the matrices must be stochastic with nonnegative elements, which has restricted the generalization of the result to more complicated situations. For instance, for the asynchronous leader-following consensus problem with an active leader, the protocol is designed to make sure that the followers should finally track the dynamic desired trajectory generated by the leader. After applying the model transformation technique, the problem can be converted into the convergence problem of the PIGSSM, where the row sums are less than or equal to 1 and also with some negative elements. Therefore, the Wolfowitz theorem is no longer applicable in dynamic leader-following consensus in asynchronous setting.

On the other side, as one of the popular related topics of cooperative control of multi-agent systems, sampled-data control is of particular interest in many practical situations. It can convert a continuous-time signal into a digital signal by utilization of digital devices, and cope with unreliable information channel, limited data sensing and transmission cases and so on. The common mathematical machinery and tools employed in synchronous sampled-data control includes nonnegative matrix theory especially the products property of infinite stochastic matrices [4], [17], characteristic equation theory [26], Lyapunov function method [27], [30], and so on. For more details, please see the survey [8]. During the implementation of sampled-data control, the systems become inevitably asynchronous when the digital devices work at different frequencies or spatially scatter in a large area without a central clock, so it is of great necessity to study the asynchronous sampled-data control. In [25], Xiao et al. established a Lyapunov-based stability result for asynchronous sampled-data multi-agent systems with time-varying delays under fixed topology. Nevertheless, because of the limitation of Lyapunov function method, it is rather difficult to construct a common Lyapunov function for asynchronous sampled-data problem under switching topology. Furthermore, in the case of asynchronous sampled-data problem with an active leader, the dynamic leader-following consensus for asynchronous sampled-data systems can be transformed into the convergence problem of the PIGSSM. Similar to the analysis in the last paragraph, the applications of Wolfowitz theorem often accompany with supplementary conditions, such as the nonnegativity of elements in stochastic matrices. Due to the existence of negative elements in the general sub-stochastic matrices, it is rather difficult or even impossible to apply the Wolfowitz theorem in the dynamic leader-following consensus problem of the asynchronous sampled-data systems.

We are thus motivated to investigate the dynamic leader-following consensus for asynchronous sampled-data multi-agent systems under switching topology. A new approach is developed to overcome the difficulties encountered by the above methods. After the implementation of specific linear transformation, the problem of dynamic leader-following consensus for asynchronous sampled-data multi-agent systems will focus on the convergence problem of the error systems, which can be resolved by presenting convergence of the PIGSSM. The general sub-stochastic matrices are matrices with row sum less than or equal to 1 but with some negative entries. We split the general sub-stochastic matrix into two different parts: (1) a sub-stochastic matrix, that is a nonnegative matrix with row sum less than or equal to 1; (2) a matrix containing negative elements and with row sum 0. Moreover, we present its property of infinite norm by a graphical approach for the sub-stochastic matrix. Finally, we utilize the property of infinity norm and some innovative zoom techniques to give the convergence analysis.

The remainder of this paper is organized as follows. Some preliminaries about graph theory and matrix theory are given in Section 2. Section 3 formulates the problem to be investigated and the main result is stated in Section 4. In Section 5, some numerical examples are demonstrated to verify the effectiveness of theory. Finally, conclusions are drawn in Section 6.