Distributed anti-windup approach for consensus tracking of second-order multi-agent systems with input saturation

Abstract

In this paper, we study the global consensus tracking problem for second-order multi-agent systems with input saturation under general directed communication graphs. A controller design approach based on distributed anti-windup control is proposed to handle the input saturation constraint. We generalize the traditional two step anti-windup control methodology to multi-agent systems and solve the input constrained consensus tracking problem based on distributed anti-windup compensator design. Some simulation examples are provided to illustrate the effectiveness of the proposed control strategy.

Introduction

In recent years, intensive research attention has been devoted to the coordination control of multi-agent systems due to its broad applications in areas such as mobile sensor network, unmanned aerial vehicles and satellite attitude alignment [1]. In distributed control of multi-agent systems, the goal is to design distributed control laws which only use local information such that some global control objectives can be achieved. Common coordination tasks include consensus, consensus tracking, flocking, swarming and so on. In the consensus tracking problem, there exists some leader agent which determines the final desired trajectory of the network and the follower agents need to track the trajectory of the leader asymptotically. Various results have been obtained for this problem regarding different agent dynamics and communication topologies [2], [3], [4], [5], [6].

When implementing the coordination controllers to real systems, practical limitations of the agent dynamics have to be considered. Input saturation is one ubiquitous control limitation imposed by physical systems resulted from finite actuation power of the agents. It may lead to serious performance degradation or even instability if not properly handled. In this sense, designing coordination controllers subject to input saturation is of great importance. In [7], [8], saturated consensus of first-order integrators with directed communication graphs was studied. Consensus algorithms for double-integrator dynamics with bounded inputs were proposed in [9]. In [10], consensus strategies accounting for actuator saturations and the lack of velocity measurements were designed for double-integrator dynamics based on auxiliary systems. [11] investigated the saturated global consensus tracking problem for neutrally stable systems and double integrator systems under detail-balanced directed communication graphs. A globally bounded nonlinear controller was proposed in [12] for second-order multi-agent systems under detail-balanced directed graphs from the input constrained optimal consensus perspective. Semi-global and global containment control were studied for second-order multi-agent systems with input saturation in [13]. In [14], [15], [16], [17], semi-global coordination of linear multi-agent systems with input saturation was studied with low-gain and low-and-high gain control approaches. Using similar techniques, semi-global consensus of linear multi-agent systems with relative output feedback and input saturation under directed switching networks was investigated in [18]. In [19], a multi-hop relay based distributed controller was proposed for global leader-following consensus of linear multi-agent systems with input saturation. Global saturated consensus problem for discrete-time neutrally stable and double integrator multi-agent systems was studied in [20]. Model predictive control methods were employed in [21], [22] to handle input saturation constraints in consensus control.

Note that for global consensus tracking of second-order multi-agent systems with input saturation, existing results are mostly only applicable for undirected or detail-balanced directed communication graphs [9], [10], [11], [13], [19]. For general directed communication graphs, the asymmetry of the graph Laplacian matrix makes it much harder to analyze the effect of saturation nonlinearity on the stability of the closed-loop systems. In this work, we overcome such difficulty by generalizing the traditional anti-windup control ideas to distributed control of multi-agent systems. Anti-windup control is an effective method to counteract the influence of input saturation [23], [24], [25] and has recently been applied to the cooperative control of multi-agent systems. In [26], the global consensus of multiple identical Hurwitz linear continuous-time systems within a fully connected network in the presence of saturation was studied. A decentralized dynamic output feedback control law was designed with an anti-windup compensation term. The results were generalized to the regional $H_{\infty }$ synchronization of general linear multi-agent systems under input saturation in [27]. In this work, we consider the global input saturated consensus tracking problem for second-order multi-agent systems under general directed communication graphs. The controllers are obtained by adapting linear consensus tracking controllers designed for the unconstrained case and the design process is both simple and effective. Furthermore, only relative output measurements are needed for the controller implementation.

The rest of the paper is organized as follows. In Section 2, some preliminaries and the problem formulation are given. In Section 3, the distributed anti-windup approach is proposed and analyzed. In Section 4, some simulation examples are provided to illustrate the proposed control strategy. Finally, some concluding remarks are given in Section 5.

Notations

For a vector $x={[x_1,\dots ,x_n]}^T$, $\mathrm{diag}\left\{x\right\}$ is a diagonal matrix with the diagonal elements $x_1,\dots ,x_n$, ${\left|x\right|}_2$ is the Euclidean norm, and $\mathrm{sgn}\left(x\right)={[\mathrm{sgn}\left(x_1\right),\dots ,\mathrm{sgn}\left(x_n\right)]}^T$ is the signum function. For a matrix $A$, ${\left|A\right|}_2$ is the induced $2$-norm and ${\lambda }_{min}\left(A\right)$ denotes the smallest eigenvalue when all the eigenvalues are real. For a signal $w\left(t\right)$ defined on $[0,+\infty )$, its $2$-norm is defined as ${\Vert w\left(t\right)\Vert }_2=\sqrt{{\int }_0^{\infty }{\left|w\left(t\right)\right|}_2^{2}dt}$. A signal $w\left(t\right)$ is said to belong to ${\mathcal{L}}_2$, i.e., $w\left(t\right)\in {\mathcal{L}}_2$, if ${\Vert w\left(t\right)\Vert }_2<\infty$.