Distributed consensus control for second-order nonlinear multi-agent systems with unknown control directions and position constraints

doi:10.1016/j.neucom.2018.03.063

Neurocomputing

Volume 306, 6 September 2018, Pages 61-67

https://doi.org/10.1016/j.neucom.2018.03.063 Get rights and content

Abstract

This paper investigates the leaderless consensus problem in the presence of unknown control directions and position constraints under directed graph. Based on the Nussbaum-gain technique and Barrier Lyapunov functions, the position-constrained consensus protocol is proposed for the multi-agent systems with unknown control directions. The proposed protocol ensures that all the signals in the closed-loop system are globally bounded and the consensus errors asymptotically converge to zero. Moreover, during the process of consensus, the trajectory of the position state of each agent is contained in the open interval which can be chosen arbitrarily in advance. A simulation example is given to demonstrate the effectiveness of the proposed control protocol.

Introduction

Apart from its theoretical significance, dealing with control systems in a distributed sense is useful and even essential in many practical scenarios such as cooperative surveillance, multiple robotics, and robot formation control [1], [2], [3]. Consensus problem, as an important research topic of distributed control, is studied how to design a distributed control protocol such that all agents can ultimately reach an agreement on some common state.

Consensus problem for multi-agent systems described by first-order integrators was extensively discussed in early years [4], [5], [6]. As for the second-order or even high-order multi-agent systems, Das and Lewis [7] designed a distributed controller for second-order multi-agent systems. Afterwards, Zhang and Lewis [8] and El-Ferik et al. [9] further generalized the result to higher-order multi-agent systems. However, as remarked in [10], [11], these controllers in [7], [8], [9] are all not fully distributed, since the controller of each agent requires the knowledge of entire communication topology which is seen to be the global information. To improve the aforementioned drawback, distributed leader–follower consensus protocols were presented in [12], [13], in which no information about the whole communication topology is required. But we note that these two papers consider only the special case of communication topologies. More specifically, they are assumed to be “no loop” in communication graph. For this reason, Li et al. [10] and Mei et al. [14] presented fully distributed control protocols under general cases. Indeed, they assume only that the communication topology contains a spanning tree with the leader as the root. As to the discrete-time linear multi-agent systems, Liu et al. [15] handled the H_∞ consensus problem for a class of linear discrete time-varying multi-agent systems with general dynamics over a finite horizon. Recently, fully distributed low-complexity control problem for strict-feedback multi-agent systems was addressed in [16], where the control protocol archives the prescribed performance of the closed-loop system.

It needs to be emphasized that the results above are all based on the assumption that the control direction of each agent can be known in advance. However, in real-life applications, the sign of input function of each agent might not always be known a priori. It may be caused by using some unknown types of devices in the multi-agent systems. For example, there are many manipulators driven by armature-controlled direct current (DC) motors for which the actuator voltage is the actual control input. However, the positive pole and negative pole of the voltage corresponding to the DC motors may not always be known a prior. It leads to the consequence that the signs of input function are unknown. In many literatures, the signs of input function are also called the control directions. Considering the consensus problem under such situations, using Nussbaum-type gain function which was first proposed in [17], is one of the main way to solve this kind of problem. Peng and Ye [18] designed a Nussbaum-type adaptive controller for first-order leaderless multi-agent systems with considering unknown control directions. Later, Chen et al. [19] presented a novel Nussbaum-type gain function to design the distributed leaderless consensus protocol under undirected graph with unknown control directions for first-order and second-order nonlinear multi-agent systems. Recently, Wang et al. [20] employed Chen’s Nussbaum-type gain function to design both leaderless and leader–follower fully distributed consensus protocols under directed graph with unknown control directions for second-order nonlinear multi-agent systems.

In practical, many dynamic systems, even if they are not multi-agent systems, satisfy some types of constraints. The constraints may be aroused by the temperature limit of chemical reactor, physical limit of mechanical systems, etc. Utilizing Barrier Lyapunov function is an easy and powerful technique in designing control protocol and analyzing stability of these systems. As to output constraints, the adaptive controllers for nonlinear strict-feedback systems with time-unvarying and time-varying output constraints were designed in [21] and [22] by means of Barrier Lyapunov functions. As to partial state constraints, Tee and Ge [23] addressed control problems for nonlinear strict-feedback systems with partial state constraints by using Barrier Lyapunov functions. As to full state constraints, Liu et al. [24] presented an adaptive tracking controller for nonlinear pure-feedback systems with full state constraints by means of Barrier Lyapunov functions. Recently, by combining Barrier Lyapunov function and Nussbaum-type gain technique, Liu et al. [25] presented an adaptive tracking controller for nonlinear strict-feedback systems with full state constraints and unknown control directions. However, all the results above are for single systems. To the best of our knowledge, there are few results for multi-agent systems which subject to state constraints. Panagou et al. [2] presented a novel class of Lyapunov-like barrier functions to design the distributed formation control protocol for multi-robot networks. Both the holonomic and nonholonomic constraints are considered in that paper. However, the paper is under proximity graph and without considering unknown control directions.

Motivated by the facts above, this paper considers distributed leaderless consensus control for second-order nonlinear multi-agent systems under directed graph with unknown control directions and partial state constraints, i.e., position constraints. In this paper, the leaderless control protocol is proposed by means of Nussbaum-type gain technique, Barrier Lyapunov functions and adaptive control technique. It is shown from the Barrier Lyapunov function that the proposed protocol ensures that all the signals in the closed-loop system are globally bounded and the consensus errors asymptotically converge to zero. Moreover, during the process of consensus, the trajectory of the position state of each agent is contained in the open interval which is chosen arbitrarily in advance. We stress that the results in [20], [21], [22], [23], [24], [25] cannot be directly used for leaderless consensus with both state constraints and unknown control directions since Lemma 1 in [20] is not available in the presence of unknown control directions and in [24] Theorem 1 cannot provide that the control errors converge to zero.

The main contributions in this paper are as follows. First, compared with the distributed consensus controllers in [19], [20], not only the unknown control directions but also the partial state constraints are considered in the design of the leaderless consensus protocol and the model of multi-agent systems under directed graphs. Second, compared with the controller in [21], [22], [23], [24], [25], we further generate the control problem with constraints and unknown control directions for single systems to the multi-agent systems under directed graph.

The following notations will be used throughout this paper: Let ‖ · ‖ denote the Euclidean norm of a vector; | · | is the absolute value of a real number; let $\bar{σ} (\cdot)$ and σ( · ) denote, respectively, the maximum and the minimum singular value of a matrix; denote by matrix A > 0 if matrix A is positive definite.

Section snippets

Graph theory

Suppose that the communication topology among N agents is represented by a directed graph $G = {V, E}$ without self-loops. Here $V$ is a finite nonempty node set where each node stands for an agent and $E \in V^{2},$ a set of ordered pairs of nodes, is the edge set. An edge $(j, i) \in E$ means that node j is a neighbor of node i, then node i can receive information from node j. Denote by $N_{i}$ the set of all neighbors of agent i. A directed path from node i₁ to node i_s is a sequence of ordered adjacent edges of the

Position-constrained leaderless consensus protocol

In this subsection, we consider the position-constrained leaderless consensus, that is, there does not exist a leader in the communication topology among N agents. Based on the relative information exchanging among the neighboring agents, the adaptive consensus control protocol is designed such that all agents achieve leaderless consensus, i.e., $\lim_{t \to \infty} (x_{i} (t) - x_{j} (t)) = 0$ and $\lim_{t \to \infty} v_{i} (t) = 0,$ $\forall i, j = 1, \dots, N$ . Moreover, during the process of consensus, the trajectory of the position state x_i(t) of each

Illustrative example

In this section, we present a simulation example to verify the effectiveness of the proposed method. The multi-agent system of this example is composed of four agents where each agent is a single-link robot. The directed graph associated with the communication topology is shown in Fig. 1. The Laplacian matrices associated with the directed graphs is given as $\begin{matrix} L = [\begin{matrix} 0.3 & 0 & - 0.1 & - 0.2 \\ - 0.3 & 0.3 & 0 & 0 \\ 0 & - 0.4 & 0.4 & 0 \\ - 0.2 & 0 & 0 & 0.2 \end{matrix}] . \end{matrix}$

As illustrated on page 140 of [30], the dynamics of ith agent, for $i = 1, 2, 3, 4,$ is given by $\begin{matrix} \begin{matrix} J_{i} {\ddot{q}}_{i} \end{matrix} \end{matrix}$

Conclusion

In this paper, the position-constrained leaderless consensus problem is given and addressed for second-order nonlinear multi-agents with unknown control directions under directed graphs. The design methods are not necessary to know the signs of input functions except for their bounds due to using the Nussbaum gain in the control design. Moreover, the position constraints can be arbitrary given provided that the initial values fulfill them.

Xuan Cai received the M.E. degree from University of Shanghai for Science and Technology, Shanghai, China in 2015 and is currently pursuing the Ph.D. degree in control science and engineering at University of Shanghai for Science and Technology, Shanghai, China. His current research interests include nonlinear control theory distributed control of nonlinear systems and adaptive control.

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Chaoli Wang (M’09) received the B.S. and M.Sc. degrees from Mathematics Department, Lanzhou University, Lanzhou, China, in 1986 and 1992, respectively, and the Ph.D. degree in control theory and engineering from the Beijing University of Aeronautics and Astronautics, Beijing, China, in 1999. He is a Professor with the School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai, China. From 1999 to 2000, he was a Post-Doctoral Research Fellow with the Robotics Laboratory of Chinese Academy of Sciences, Shenyang, China. From 2001 to 2002, he was a Research Associate with the Department of Automation and Computer-Aided Engineering, the Chinese University of Hong Kong, Hong Kong. Since 2003, he has been with the Department of Electrical Engineering, University of Shanghai for Science and Technology, Shanghai, China. His current research interests include nonlinear control, robust control, robot dynamic and control, visual servoing feedback control, and pattern identification.

Gang Wang was born in Chifeng, China, in 1990. He received the B.Sc. degree in Information and Computing Science and the Ph.D. degree in Systems Analysis and Integration from University of Shanghai for Science and Technology, Shanghai, China, in 2012 and 2017, respectively. He is currently a Research Associate in the Department of Electrical and Biomedical Engineering, University of Nevada, Reno. His research interests include distributed control of nonlinear systems, adaptive control, and robotics.

Dengyu Liang received the B.Sc. degree in Mathematics and Applied Mathematics from Hebei Normal University, China, in 2015. He is currently pursuing the Ph.D. degree in control science and engineering with University of Shanghai for Science and Technology. His current research interests include the control of nonholonomic mobile robots and adaptive control.

^☆: This paper was partially supported by the National Natural Science Foundation of China (61374040, 61673277, 61503262).

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Distributed consensus control for second-order nonlinear multi-agent systems with unknown control directions and position constraints☆

Abstract

Introduction

Section snippets

Graph theory

Position-constrained leaderless consensus protocol

Illustrative example

Conclusion

Automatica

Automatica

IEEE Trans. Syst. Man Cybern.: Syst.

Syst. Control Lett.

Syst. Control Lett.

J. Frankl. Inst.

Automatica

Automatica

Automatica

Automatica

Multiple UAV coalitions for a search and prosecute mission

J. Intell. Robot. Syst.

Distributed coordination control for multi-robot networks using Lyapunov-like barrier functions

IEEE Trans. Autom. Control

Distributed leaderless consensus algorithms for networked Euler Lagrange systems

Int. J. Control

Coordination of groups of mobile autonomous agents using nearest neighbor rules

IEEE Trans. Autom. Control

Consensus problems in networks of agents with switching topology and time-delays

IEEE Trans. Autom. Control