Robust adaptive distributed dynamic surface consensus tracking control for nonlinear multi-agent systems with dynamic uncertainties

Abstract

This paper proposes an adaptive distributed consensus tracking control approach for uncertain nonlinear multi-agent systems in pure-feedback form under a directed topology where each follower is dominated by dynamic uncertainties and unmeasured states. Radial basis function neural networks (RBFNNs) are employed to compensate the unknown nonlinear functions obtained by recursive design procedure for followers. The distributed dynamic surface controllers are able to eliminate the condition in which the approximation error of the traditional neural networks is bounded. By introducing an available dynamic signal and two smooth scalar functions, the obstacle caused by unmodeled dynamics is conquered. The main advantage of the proposed method is that for M pure-feedback nonlinear followers, only one learning parameter needs to be updated online. It is also shown that the proposed consensus controller can guarantee cooperatively semi-global uniform ultimate boundedness (CSUUB) of all the signals, and the consensus errors converge to an adjustable neighborhood of the origin.

Introduction

Research on distributed control of networked multi-agent systems has attracted tremendous attention due to their applications in formation control [19], [2], event-triggered consensus control [3], smart microgrid energy management [6], containment control [14], energy internet [22], distributed estimation [7], [17] and other areas. The consensus problem means to guarantee the states of a group of networked agents to reach an agreement. An important consensus-like problem is called the distributed consensus tracking control, where all the followers are trying to follow the leader, and only a subset of the networked group can get information from a time-varying leader. Numerous results concerning multi-agent distributed tracking control have been published in recent years [4], [21], [12], [15], [16], [5], [33], [20], [35], [18]. In [4], a consensus tracking protocol was given for multi-agent systems with continuous single-integrator dynamics, which had measurement noises and time-delays. In [21], the authors studied leader-following consensus problem for double-integrator multi-agent systems with velocity measurements based on pining control. The work is extended in [15] using only relative position measurements and in [16] to a coordinated tracking algorithm combined with a distributed filter. For more practical purposes, the consensus tracking problems of multi-agent systems were investigated for high-order nonlinear systems [5], [33], [20] and multiple rigid bodies [35], [18]. It should be pointed out that most of these aforementioned studies [4], [21], [15], [16], [5], [33], [20], [35], [18] only deal with nonlinear multi-agent systems with matched uncertainties. However, various physical systems have more complicated dynamics without satisfying the matching conditions.

On the other hand, adaptive backstepping technique has become a powerful tool to handle strict-feedback nonlinear systems without matching condition. In [11], adaptive backstepping controller was first designed for parametric strict-feedback systems with over-parametrization to obtain asymptotic tracking performance. By introducing universal function approximators, the systematic and recursive control method was further extended and applied to a large class of SISO nonlinear systems with unstructured uncertainties [1], [26], [30]. Dynamic surface control method [23], [28] has been proven to be useful for solving the problem of explosion of complexity in the traditional adaptive neural or fuzzy network based on backstepping design. Recently, dynamic surface control approach has been improved to control nonlinear multi-agent systems [32], [31]. In [32], a neural network-based distributed consensus tracking problem for uncertain multi-agent strict-feedback systems was studied by using the dynamic surface control technique, in the case where only a small fraction of follows has access to a time-varying leader, while in [31], based on the Lyapunov׳s stability theory, a distributed adaptive containment control approach was proposed for uncertain nonlinear multi-agent systems in strict-feedback form by employing neural networks. It was pointed out that the agents in [32], [31] were modeled by strict-feedback systems, and they had affine appearance of the state variables.

Unmodeled dynamics and dynamic disturbances usually co-exist in various practical systems. Their existence often have serious effects on the performance and the stability of the control systems. In past decades, a great number of works have been done for several types of nonlinear systems to address this problem based on the adaptive backstepping technique. In [8], [9], two adaptive backstepping control methods were proposed for a class of parametric strict-feedback nonlinear systems with unmeasured input-to-state stable dynamics. The robustness of the unmodeled dynamics and dynamic disturbances is ensured by introducing a Lyapunov function. In [24], a stable adaptive fuzzy robust approach was proposed for strict-feedback canonical nonlinear systems with unmodeled dynamics by combining fuzzy logic systems with the small-gain approach; while in [25], a direct fuzzy adaptive control method was developed for a class of SISO nonlinear systems with completely unknown virtual control directions. By integrating changing supply function with dynamic surface control technique, a new robust adaptive output feedback control scheme was investigated based on a fuzzy state observe in [34]. In [27], an adaptive observe-based neural control problem was discussed for a class of stochastic nonlinear strict-feedback systems with unmodeled dynamics and dead-zone input by using a stochastic small-gain theorem. The authors in [24], [25], [34], [27] use a dynamic signal to compensate for the unmodeled dynamics. However, it is assumed that the upper bound functions with respect to the dynamical disturbance in [8], [9], [24], [25], [34], [27] are known, which is very crucial in the controller design.

Motivated by the above observations, we consider the distributed tracking control problem for uncertain pure-feedback multi-agent systems with unmodeled dynamics and dynamic disturbance. The main contributions of the proposed control approach in this paper lie in:

(i) Different from the traditional adaptive fuzzy or NN backstepping controllers designed for individual agent system [11], [1], [26], [30], [23], [28], a novel distributed tracking control scheme for nonlinear multi-agent systems needs to consider the coupled terms in dynamics, the communication among the agents, and the ability on information transmission of a time-varying leader and so on. By utilizing dynamic surface control technique, the possible circular argument is avoided in Lyapunov stability analysis.

(ii) Each follower node is a pure-feedback integrator incorporated with dynamic uncertainties and unmodeled dynamics. Most of the existing results described the dynamics of nonlinear multi-agent systems with unmatched uncertainties as strict-feedback form [32], [31]. Recently, distributed consensus tracking control problem was investigated for nonlinear pure-feedback multi-agent systems in [29], in which the partial derivatives $\frac{\partial f_{i,k}({\overline{x}}_{i,k},x_{i,k+1})}{\partial x_{i,k+1}}$ are strictly assumed to be either positive or negative. However, our approach no longer needs such a restrictive assumption. Besides, unlike in [29], we consider unmodeled dynamics which makes the system model much more complex and practical.

(iii) In contrast to the distributed consensus tracking problem for multi-agent systems with uncertain nonlinearities based on adaptive neural networks or fuzzy logic systems [5], [20], [29], [31], [32], [33], only one parameter is tuned on line regardless of the weights of neural networks.

The rest of the paper is organized as follows. Section 2 formulates the problem. In Section 3, distributed dynamic surface control is proposed for a class of nonlinear multi-agent systems with M followers and a leader under a directed network, then the stability of the closed-loop system is given. Simulation results are performed to demonstrate the effectiveness of the proposed control scheme in Section 4, and Section 5 concludes the paper.