Leader-following consensus of data-sampled multi-agent systems with stochastic switching topologies

Abstract

This paper investigates the leader-following consensus problem for multi-agent systems with Markovian switching topologies in a sampled-data setting. We study two algorithms corresponding to the case where the leader׳s state is time varying or time invariant. With a time-varying leader׳s state, we present necessary and sufficient conditions for boundedness of the tracking error systems. With a time-invariant leader׳s state, we present necessary and sufficient conditions for mean-square stability of the tracking error systems. An optimization algorithm is given to derive the allowable control gains or the feasible sampling period. Simulation examples are presented to show the usefulness of the results.

Introduction

Leader-following consensus has received significant attention in the control field in recent years. This is mainly due to its wide applications in engineering, such as unmanned air vehicles, mobile robotic swarms, wireless sensor networks and cooperative surveillance [1], [2], [3], [4], [5]. The main idea of leader-following consensus is that the leader sends its state information to the followers directly or indirectly such that the tracking errors between leader and all followers are as small as possible. There are many publications on the topic of the leader-following consensus problem [6], [7], [8]. In [6], the multi-agent system considered was with measurement noises and directed interconnection topology. A sufficient condition for mean-square stability of the closed loop tracking control system was obtained by designing distributed estimators. In [7], both leaderless and leader-following consensus problems were studied. The stability or boundedness conditions were presented based on Lyapunov theorems and Nyquist stability criterion. By using the sampled-data control approach, the leader-following consensus for multi-agent systems was studied in [8]. The topology considered is deterministic.

Data-sampled approach is frequently used to discretize the continuous-time system in control community. In recent years this method is also used to study the multi-agent systems [8], [9], [10], [11], [12]. Two sampled-data coordination algorithms for double-integrator dynamics were studied in [10] where the interaction topology is fixed undirected/directed. In [11], the consensus problem of double-integrator multi-agent systems with both fixed and switching topologies was studied. The switching signal is arbitrary and only a sufficient condition is derived to solve a consensus problem in this case. In [12], the authors researched the stochastic bounded consensus tracking problems of multi-agent systems, where the sampling delay induced by the sampling process was considered.

The topologies in the above literature are all deterministic or switching in a deterministic framework. However, the system models are sometimes switching stochastically due to the internal or/and external disturbance. Similar to some other control systems, the Markovian switching model has been used to describe the interaction topology among the agents in very recent years [13], [16]. In [13], the static stabilization problem of a decentralized discrete-time single-integrator network with Markovian switching topologies was studied. In [14], the authors considered the consensus for a network of single-integrator agents with Markovian switching topologies. In [15], the authors studied the mean-square consentability problem for a network of double-integrator agents with Markovian switching topologies. In [16], the authors studied the distributed discrete-time coordinated tracking problem for multi-agent systems with Markovian switching topologies in case of the transition probabilities are equal.

Motivated by the former considerations, we will extend the leader-following consensus problem in [8] to the case of Markovian switching topologies in this paper. In this case, the leader-following consensus problem will become more challenging. Both time-invariant and time-varying leader are considered. Based on algebra graph theory and Markovian jump system theory, we present the necessary and sufficient conditions for the convergence of the tracking error systems. An optimization algorithm will be given to derive the allowable control gains and sampling period.

Notation: Let $\mathbb{R}$ and $\mathbb{N}$ represent, respectively, the real number set and the non-negative integer set. Denote the spectral radius of the matrix M by $\rho \left(M\right)$. Suppose that $A,B\in {\mathbb{R}}^{p\times p}$. Let $A\succcurlyeq B$ (respectively, $A\succ B$) denote that $A-B$ is symmetric positive semi-definite (respectively, symmetric positive definite). Denote the determinant of the matrix A by $\parallel A\parallel$. Given $X\left(k\right)\in {\mathbb{R}}^p$, define ${\parallel X\left(k\right)\parallel }_E\triangleq {\parallel E\left[X\right(k\left)X^T\right(k\left)\right]\parallel }_2$, where $E[·]$ is the mathematical expectation. “⊗” represents the Kronecker product of matrices. I_n denotes the $n\times n$ identity matrix. Let ${\mathbf{1}}_n$ and ${\mathbf{0}}_{m\times n}$ denote, respectively, the $n\times 1$ column vector with all components equal to 1 and $m\times n$ zero matrix.