Distributed robust event-triggered adaptive control for a class of uncertain nonlinear multi-agent systems with actuator saturation

Abstract

This paper studies the event-triggered consensus control problem for high-order uncertain nonlinear multi-agent systems with actuator saturation. By using a smooth Lipschitz function to approximate the saturation nonlinearity, an augment system and the Nussbaum function are adopted to deal with the residual terms of saturation nonlinearity based on adaptive backstepping method. Since excessive energy and communication resources will be consumed during the procedure to handle actuator saturation, two event-triggered mechanisms are proposed to save the communication resources and reduce the controllers’ update frequency. Whenever the triggered conditions are satisfied, the control signals transmitted to the actuators are updated and broadcasted to the neighboring area. A ’disturbance-like’ term is integrated so that the event-triggered control problem with actuator saturation can be transformed into a robust problem while the unknown disturbances are tackled by adaptive update laws. Moreover, the requirement for global communication topology known by all the agents is relaxed by introducing new estimators. All the signals in the closed-loop system are uniformly bounded and the consensus tracking errors are exponentially converged to a bounded set. Meanwhile, the Zeno behavior is excluded. Simulation results are employed to validate the advantages of our proposed methods.

Introduction

Consensus control for Multi-Agent Systems (MASs) has a wide range of applications and received extensive attention in various fields, including mobile robotics, flight vehicles, and smart grids [1], [2], [3]. The general structure for distributed consensus control algorithm is given in [1], then several control approaches have been proposed to flourish this area [4], [5], [6], [7], [8]. Most of the early results require the system model to be linear and precise. However, the widespread nonlinearities and uncertainties have restricted the applications of the early consensus control strategies. Therefore, it’s practical to investigate the consensus control problems for MASs with the nonlinearity, the intrinsic mismatched parameters, and the external disturbances. A series of articles have studied the distributed consensus control problem for nonlinear MASs, in which the adaptive backstepping method has been proved to be powerful for dealing with the uncertain nonlinear dynamics [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. In [9], the asymptotical consensus tracking for nonlinear MASs with intrinsic mismatched unknown parameters is achieved. Then, in [12], the unknown external disturbances are well handled by introducing extra estimators. In [13], the neural network-based state observer is constructed with backstepping method to handle the unmeasurable state problem of uncertain nonlinear MASs. Except for the uncertainty in the dynamics of agents, the assumption that all the agents can access the global information of the interaction graph also increased the difficulty to achieve consensus for MASs. In [20], this assumption is relaxed by introducing new estimators and error variables for first-order nonlinear MASs without disturbances.

Due to the nonlinear parts, not to mention the complicated effect by the environment, the magnitude of the designed control signal is likely to be so high that leads to the actuator saturation in practice, which can severely affect the performance of our control scheme, even disrupt the stability of the control system. For MASs, the problem about input saturation has been considered for linear dynamic cases [21], [22], [23], [24] and nonlinear dynamic cases [25], [26], [27], [28], [29], [30]. The cooperative tracking is achieved for nonlinear MASs under input constraints by system transformation in [26]. For high-order nonlinear MASs under input saturation, the situation without disturbances is investigated in [29], [30]. Furthermore, the generalized Nyquist method is first employed to establish the consensus criterion under the condition of actuator saturation in [31].

In practice, it is supposed to be an embedded microprocessor in each agent to collect the information and generate the control signal to drive the agent, of which the energy and communication resources are limited. Event-triggered control mechanism has been verified to be effective to reduce the excessive communication burden, since the information transmission occurs only when the designed conditions are satisfied. In view of this, many researchers have investigated event-triggered control with various dynamics and different triggering mechanisms [19], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43]. Event-triggered control methods are purposed for linear dynamics in [33], [35], which are robust to external disturbances. The requirement of the global Laplacian information is relaxed for linear MASs in [36]. Asymptotical tracking is achieved for first-order nonlinear MASs with triggered communication in [37]. The distributed event-triggered control methods are proposed for high-order nonlinear MASs in strict-feedback form [39] and nonstrict-feedback form [34]. Since the methods above [21], [22], [23], [24], [25], [26], [27], [28], [29], [30] have increased the computation and communication burden when handling the saturation parts, it’s necessary to adopt the event-triggered mechanisms to reduce the excessive load. However, the event-triggered consensus control problem under actuator saturation is not fully addressed [44], [45], [46], [47], [48]. For nonlinear MASs with actuator saturation, event-triggered adaptive controllers are constructed by a new dynamic output feedback method [44], and the semi-global consensus is achieved by a fully distributed event-triggered approach [45]. In [47], an adaptive neural network method is developed for leader-following nonlinear MASs with input saturation and unknown disturbances on the directed graph.

Inspired by the above results, this paper addresses the distributed consensus control problem for nonlinear MASs under event-triggered mechanisms with only locally available information, each agent is modeled as the high-order uncertain nonlinear dynamic with actuator saturation. By utilizing the adaptive backstepping method, Nussbaum function, and ”disturbance-like” term, we propose two new event-triggered control laws that can achieve the global stabilization of such agents on the directed graph and exclude the Zeno behavior. The main contributions of this paper are summarized as follows.

• The event-triggered consensus control problem for nonlinear uncertain MASs with actuator saturation under the directed graph is investigated. Compared with the results above, the dynamics of agents are more general with high-order strict-feedback nonlinear dynamics, unknown parameters and external disturbances for each state channel, which makes our control problem more practical and challenging.

• The information transmission of the control signal is based on the designed event-triggered mechanisms, which take the saturation nonlinearity into consideration. It has better performance for reducing the communication burden than the relative and fixed threshold strategies. The triggering error caused is easily tackled by introducing the ’disturbance-like’ term.

• The amount of adaptive estimators is reduced compared with [12], [28], and the disturbance terms coupled are well handled by introducing new adaptive update laws in each step. Meanwhile, the global information of the communication graph is no longer required by introducing adaptive update laws for the unknown parameters existed in the topology matrix.

This paper is outlined as follows. We formulate the distributed event-based consensus control problem for nonlinear MASs in Section 2. The control signals and adaptive update laws are designed in Section 3. Sections 4 and 5 show the stability analysis and simulation results respectively. Finally, conclusions are given in Section 6.

Notation: In this paper, $N$ represents the number of agents. $\parallel .\parallel$ denotes the Euclidean Norm. $\mathfrak{R}$ means the n-dimensional Euclidean space. $diag(\cdots )$ is the diagonal matrix. $\widehat{\Delta }$ means the estimate for $\Delta$, and $\tilde{\Delta }=\Delta -\widehat{\Delta }$ stands for the estimation error. $B\left(\Delta \right)$ is the bound of $\Delta$.