Adaptive iterative learning protocol design for nonlinear multi-agent systems with unknown control direction

doi:10.1016/j.jfranklin.2018.04.012

Journal of the Franklin Institute

Volume 355, Issue 10, July 2018, Pages 4298-4314

https://doi.org/10.1016/j.jfranklin.2018.04.012 Get rights and content

Abstract

This paper investigates a new adaptive iterative learning control protocol design for uncertain nonlinear multi-agent systems with unknown gain signs. Based on Nussbaum gain, adaptive iterative learning control protocols are designed for each follower agent and the adaptive laws depend on the information available from the agents in the neighbourhood. The proper protocols guarantee each follower agent track the leader perfectly on the finite time interval and the Nussbaum-type item can seek control direction adaptively. Furthermore, the formation problem is studied as an extension. Finally, simulation examples are given to demonstrate the effectiveness of the proposed method in this article.

Introduction

Coordination and cooperative control of multi-agent systems has received a great amount of attentions due to its broad applications in various areas [1], [2], [3], [4], [5], [6], [7]. Consensus is a fundamental problem for cooperative control, and the basic idea of consensus control is that all agents are driven to an agreement by a consensus protocol. It is well known that Nussbaum gains can be used to tackle adaptive control with unknown control gains for nonlinear systems. In literature [8], the authors solved the adaptive consensus problem for first and second order linearly parameterized multi-agent systems with unknown identical control directions, and designed a new Nussbaum-type function to solve this consensus problem. Furthermore, the author in [9] has proposed another new kind of Nussbaum gains that can be used for adaptive consensus for multi-agent systems.

Iterative learning control has been widely used to handle the repeated tracking control and the high precision tracking performance can be achieved on the finite time interval [10]. Therefore, iterative learning control has been applied to multi-agent systems. Many literatures [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23] have reported the consensus and formation problems for multi-agent systems by using iterative learning control. The quantized iterative learning problem of digital networks has been investigated in [17], [18]. The authors designed adaptive iterative learning control protocols for coordination problem of first-order and second-order multi-agent systems in [19], [20]. In [21], [22], the authors investigated the cooperative learning problems for formation control of nonlinear multi-agent agents. However, the control input gain in these works is totally known. Thus, in [23], an adaptive iterative learning control scheme for high-order nonlinear multi-agent systems was studied, where the control input gain functions is not fully known. When the control input gain is fully unknown, that is to say, a control direction for multi-agent system is not known a priori, the available approaches cannot address the perfect tracking consensus problems for this kind multi-agent system. How to design the adaptive iterative learning protocol for nonlinear multi-agent systems remains an open problem for further studies.

An effective tool to handle unknown control direction is to incorporate the technique of Nussbaum-type gains into the control design. To deal with the tracking problem without a priori knowledge of the control direction, the authors in [24] incorporated the Nussbaum-type function into the learning control design. By incorporating a Nussbaum-type function, the iterative learning control problem have been studied in [25] for nonlinear system with unknown time-varying parameters and uncertain control direction. It is noted that the results in [24], [25] only studied the problem for a single nonlinear dynamical system. However, for multi-agent systems where each agent has an unknown control direction, the analysis becomes more difficult since the multi-agent systems may have multiple Nussbaum-type function. Furthermore, the information of the agents from its neighbourhood can be used in the protocol design and all agents with unknown control directions should cooperate together to achieve consensus or form the desired formation. Thus, we need to further develop a distributed adaptive iterative leaning control law utilizing the Nussbaum gain technique for consensus and formation problems.

In this paper, we will investigate the adaptive iterative learning control design for tracking consensus and formation control problems of nonlinear multi-agent systems without a priori knowledge of the control direction. We are able to deal with uncertain nonlinear multi-agent systems, where the dynamic of each follower without knowing the control direction (both sign and amplitude unknown) and with time varying parametric uncertainties. Then, based on Nussbaum gain, a new adaptive iterative learning control scheme is investigated for multi-agent systems to ensure that only the information available in the local neighbourhood. The designed protocols guarantee each follower agent track the leader perfectly on the finite time interval and the Nussbaum-type item can seek control direction adaptively. Furthermore, as an extension of the former result, the formation problem is studied.

The remainder of this paper is organized as follows. In Section 2, the graph theory and problem formulation are given. The main results of adaptive iterative learning control scheme based on Nussbaum gains for multi-agent consensus and formation problems are proposed in Section 3. In Section 4, the effectiveness of the distributed adaptive iterative learning control algorithms proposed in this paper is demonstrated by the simulations. Finally, Section 5 gives conclusions of this paper.

Section snippets

Graph theory

Let $G = (V, E, A)$ denote an undirected graph, where $V = {v_{1}, \dots, v_{n}}$ denotes the set of vertices and E⊆V × V is the set of edges. $A = [a_{i j}] \in R^{n \times n}$ is the weighted adjacency matrix of the graph $G$ . If there is an edge between agent i and j, i.e., (v_j, v_i) ∈ E, then $a_{i j} = a_{j i} > 0,$ and otherwise $a_{i j} = a_{j i} = 0$ . Moreover, $a_{i i} = 0$ . The set of neighbors of node v_i is $N_{i} = {v_{j} : (v_{j}, v_{i}) \in E}$ . The Laplacian matrix of digraph $G$ is $L = D - A,$ where $D = diag {d_{1}, \dots, d_{n}},$ $d_{i} = \sum_{j = 1}^{n} a_{i j}$ . The graph $G$ is connected if there is a path between any two

Consensus problem for the multi-agent systems

At the kth iteration, the error dynamics is given by $\begin{matrix} {\dot{e}}^{k} = - H ({\dot{x}}^{k} - 1_{n} {\dot{x}}_{0}) = - H (ξ^{k} Θ^{k} + ρ u^{k} - 1_{n} f (x_{0}, t)) \end{matrix}$ where $ξ^{k} = diag {ξ_{1}^{k}, \dots, ξ_{n}^{k}},$ $ρ = diag {ρ_{1}, \dots, ρ_{n}},$ $u^{k} = {[u_{1}^{k}, \dots, u_{n}^{k}]}^{T}$ and $Θ^{k} = {[θ_{1}^{k}, \dots, θ_{n}^{k}]}^{T}$ .

In order to design the control protocol, in the following, a definition of convergent series sequence and its related properties are given.

Definition 1

[19]: A convergent series sequence {Δ_k} is defined as $Δ_{k} = \frac{a}{k^{l}}$ where a, l and $k \in Z_{+}$ are constant parameters to be designed and $l (\in Z_{+}) \geq 2,$ a( ∈ R) > 0. In addition, we let $Δ_{0} = 1$ in this paper.

Lemma 4

Simulations

In this section, two examples will be given to illustrate the efficiency of the new distributed adaptive iterative learning protocols proposed in this paper. The topology graph for one leader and four follower agents is shown in Fig. 1 It is obvious that the graph $\bar{G}$ is connected. The weighted adjacency matrices are $A = (\begin{matrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix})$ and $B = diag {1, 0, 0, 1}$ . The parameters for the dynamics of four follower agents are $θ_{i} (t) = 1 + \sin (i π t),$ $ξ_{i} = \cos (x_{i}^{2}),$ $i = 1, 2, 3, 4$ ; $ρ_{1} = - 4,$ $ρ_{2} = - 6,$ $ρ_{3} = - 2,$ $ρ_{4} = - 5$ . The

Conclusion

This paper deals with the cooperative problems for uncertain nonlinear multi-agent systems with unknown parameter and control direction in the finite time interval. The distributed learning control protocols based on Nussbaum gain have been proposed for each agent and the adaptive laws depend on the information available from the agents in the neighbourhood. Then, under the new adaptive iterative learning control protocols, all follower agents can follow the leader pefectly on [0, T] and the

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grants 61603286 and 61573013, the Fundamental Research Funds for the Central Universities (XJS15065, JB160702, JBG160707, XJS18012) and General Financial Grant from the China Postdoctoral Science Foundation(2015M582617).

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Adaptive iterative learning protocol design for nonlinear multi-agent systems with unknown control direction

Abstract

Introduction

Section snippets

Graph theory

Consensus problem for the multi-agent systems

Simulations

Conclusion

Acknowledgments

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