Bipartite consensus tracking problem of networked Lagrangian system with intermittent interactions

Abstract

This paper studies the bipartite consensus tracking (BCT) problem of the networked Lagrangian system (NLS) with intermittent interactions, in which the interaction among the individuals is on in the interactive time intervals and is off in the un-interactive ones. Besides, different from the existing works, where the dynamics of the system is linear or nonlinear, we consider the Lagrangian system in this paper with dynamical characteristics: high nonlinearity and coupling. In such case, a hierarchical intermittent-interactions-based control (HIIC) algorithm, including the distributed intermittent estimator and local control algorithm, is designed to achieve the above-mentioned control goal. Specifically, the distributed intermittent estimator is constructed to estimate the information of the leader for each individual. The local control algorithm is designed based on the derived estimators to address the BCT problem finally. Furthermore, the sufficient conditions for ensuring the stability of the closed-loop system are derived through systematic Lyapunov stability analysis. Finally, some numerical simulations on the networked manipulators are performed to prove the validity of the proposed HIIC algorithm.

Introduction

In recent years, the control problem of nonlinear Lagrangian systems has attracted much attention of many researchers, due to its wide applications of many aspects, such as unmanned aerial vehicles [1], marine surface vehicles [2] and robots [3], [4], etc. In practice, networked Lagrangian systems (NLSs) usually has better performance than a single individual. This is mainly due to that the NLSs have higher efficiency, stronger fault tolerance and better robustness compared with a single agent [5], [6], [7], [8]. For this reason, it motivates a group of researches on the NLSs which are connected by a distributed interactions network in many fields [9], [10], [11], [12], [13]. To meet the practical applications, the cooperative control problem of the NLSs deserves further investigations.

Among the cooperative control problems, the consensus problem is the most fundamental one, which the control objective is that if all the individuals’ states are forced to reach a uniform constant value corresponding to the initial conditions [14], [15]. Based on the results within the scope of consensus, different kinds of cooperation problems, including cluster consensus [16], [17], synchronization [18], consensus tracking [13], [19], [20], formation tracking [21] and flocking [22], have been widely studied and become a magnet in the control area. Most of the above literatures mainly focused on the cases that only cooperation among the networked systems is considered. However, in some actual application scenarios, both the cooperation and competition exist, and the NLSs are required to be split into two subgroups to perform the tasks in two opposite directions. It then motivates the generation of the researches on bipartite consensus [10], [23], [24] and bipartite consensus tracking (BCT) [25], [26], [27], [28], [29]. As for the NLSs, BCT problem is a typical application and is more in line with the actual task requirements. Considering that there are few relevant researches about the BCT problem of the NLSs, it thus becomes significant to develop novel distributed control algorithm to regulate the NLSs with both cooperation and competition.

On the other hand, most of studies on the cooperative control of the NLSs are developed under the precondition that the messages are delivered continuously among the Lagrangian agents [12], [30], [31], [32]. But in some cases, due to the limitations of sensing performance, the data packet losses and the equipment breakdown, the continuous interactions among the NLSs become unrealistic and difficult. In this cases, the individuals in the NLSs can receive the information of their neighbors in a intermittent way, where the interaction time period is divided into the interactive time interval and the un-interactive time interval. At present, related researches on the networked systems with the intermittent interactions have been wildly studied [33], [34], [35]. However, the above literatures mainly focused on second-order multi-agent system, high-order multi-agent system and other nonlinear systems. It cannot be easily extended to controller design for the NLSs, since its complex model construction, high nonlinearity and strong couplings. Besides, there are few studies on the cooperative control with intermittent interactions for the NLSs, let alone the BCT problem. Therefore, it is extremely challenging and meaningful to study the BCT problem of the NLSs with intermittent interactions.

Based on the above discussions, this paper mainly focuses on developing novel cooperative controller design for the NLS with intermittent interactions to solve the BCT problem. The main contributions are three aspects.

• A hierarchical framework, the HIIC algorithm, involving the distributed intermittent estimator and the local control algorithm, is newly designed. The BCT problem of the NLS is successfully addressed by using the above algorithm.

• Compared with [25], [26], [27], [28], where the bipartite consensus of linear multi-agent system with the continuous interactions is considered, this paper considers the BCT problem of the NLS in the case of intermittent interactions, which then is more in line with the real task requirements, and can decrease the interactions costs among the NLS.

• The hierarchical design method of the proposed algorithm can be extended to regulate other complicated networked systems with intermittent interactions, including networked marine surface vehicles, networked manipulators and networked mobile vehicles.

Notations. ${\mathbb{R}}^p$ and ${\mathbb{R}}^{p\times p}$ are the sets of the $p\times 1$ vectors and the $p\times p$ real matrix. ${\mathbf{1}}_p$ stands for the $p\times 1$ column vector with the whole elements being 1. All elements of the column vector $\mathbf{0}$ with proper dimension are 0. ${\parallel ·\parallel }_1$, ${\parallel ·\parallel }_2$ and ${\parallel ·\parallel }_{\infty }$ represent the 1-norm, the 2-norm and the $\infty$-norm, respectively. $\otimes$ represents the Kronecker product. $\text{col}(·)$ denotes the column transformation, for $m,n\in {\mathbb{R}}^p$, $\text{col}(m,n)={[m^T,n^T]}^T$. ${\lambda }_{\min }(·)$ / ${\lambda }_{\max }(·)$ are the minimum / maximum eigenvalues. $\text{sign}(·)$ is the symbolic function.