Leader-following almost output consensus for linear multi-agent systems with disturbance-affected unstable zero dynamics

Abstract

In this paper, we formulate and investigate the leader-following almost output consensus problem for linear multi-agent systems with disturbance affected unstable zero dynamics. Conditions on the communication topology, agent dynamics and the way the disturbances affect the zero dynamics are established under which low-and-high gain based consensus protocols are designed. These protocols are shown to achieve leader-following almost output consensus, that is, output consensus of the system can be achieved to an arbitrary level of accuracy with the states remain bounded in the absence of the disturbances, and when the system is operating in output consensus within the desired accuracy, the $L_2$-gain from the disturbances to the difference between each follower agent’s output with and without the disturbances from the same initial condition can be made arbitrarily small. A numerical example is shown to verify the theoretical results.

Introduction

For over a decade, the design of distributed consensus protocols for multi-agent systems has been one of the fundamental topics in distributed control (see, for example [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]). These consensus protocols are designed to achieve either leaderless consensus, where the states (outputs) of all agents converge to the same value, or leader following consensus, where the states (outputs) of all follower agents converge to the state (output) of the leader agent.

In the leader-following output consensus scenario, most of the consensus protocols designs have implicitly assumed that the agent dynamics to be minimum phase systems and not are affected by disturbances. However, many real world systems are of nonminimum phase [17]. For example, the inverted pendulum on a cart [18], and the V/STOL aircraft [19] are both nonminimum phase systems. Such nonminimum phase property of a system may prevent certain control objectives from being achieved especially if the unstable zero dynamics is affected by disturbances. Therefore, we are motivated to consider the consensus problem of multi-agent systems whose agent dynamics is of nonminimum phase and affected by the disturbance, which, to the best of our knowledge, is an open research problem.

In this paper we design consensus protocols for a class of linear multi-agent systems with disturbance-affected unstable zero dynamics. Our approach is motivated by the results on the disturbance decoupling problem for individual systems (see, for example [20], [21], [22], [23], [24]). The zero dynamics is allowed to be heterogeneous with all the poles on the $j\omega$-axis. The condition on the way the disturbance affects the zero dynamics of each follower agent is identified. The leader agent’s output to be followed can be any bounded signal that does not contain the frequency components of the $j\omega$-axis invariant zeros of the follower agents. Novel consensus protocols are constructed of a low-and-high gain feedback structure in which the low gain feedback design technique [24] is utilized to stabilize the zero dynamics of each follower agent by allowing its output to vary within a small neighborhood of the desired output. We show that, when the communication topology among the follower agents is undirected and connected, and there is a communication link between the leader agent and at least one follower agent, these protocols achieve leader-following almost output consensus, that is, the leader-following output consensus is achieved to any pre-specified degree of accuracy while the states remain bounded in the absence of the disturbances, and when the system is operating in output consensus within the desired level of accuracy, the $L_2$-gain from the disturbances to the difference between each follower agent’s output with and without the disturbances from the same initial condition is attenuated to any desired level of accuracy. The protocols can be easily generalized for agents with invariant zeros on the closed left-half plane by a nonsingular transformation (see, for example, [25]) that decompose the zero dynamics into two subsystems, one with stable poles and the other with poles on the $j\omega$-axis. We note that, compared to the results on individual systems, where the output converges toward zero precisely in the absence of the disturbances, the outputs of the multiple agents under our proposed protocols reach consensus to a time-varying desired signal only with a pre-specified accuracy. In order to analyze the effect of the disturbance on the output consensus of the system, we have to compare the output in the absence and in the presence of the disturbance.

Organization of the paper: Section 2 recalls preliminaries in graph theory. Section 3 formulates the leader-following almost output consensus problem for linear multi-agent systems with disturbances-affected unstable zero dynamics. Section 4 presents the consensus protocols as well as the analysis of the closed-loop system. Section 5 contains simulation results that verify the theoretical conclusions. Section 6 concludes the paper.

Let ${\mathbb{R}}^m$ (${\mathbb{R}}^{m\times n}$) denote the $m$ ($m\times n$) dimensional Euclidean space. For $v={(v_1,v_2,\dots ,v_m)}^{\mathrm{T}}\in {\mathbb{R}}^m$, $\Vert v\Vert =\sqrt{{\sum }_{k=1}^m{v}_k^2}$ denotes its norm. Let $\mathbf{1}$ denote ${(1,1,\dots ,1)}^{\mathrm{T}}$ and $\mathbf{0}$ denote ${(0,0,\dots ,0)}^{\mathrm{T}}$ of appropriate dimensions. Let $I_n$ ($I$) denote the identity matrix of dimensions $n\times n$ (appropriate dimensions). For a matrix $A$, $\mathrm{Im}\left(A\right)$ denotes its image, $\Vert A\Vert$ denotes its norm and $\lambda \left(A\right)$ denotes its eigenvalues. For a symmetric matrix $M\in {\mathbb{R}}^{m\times m}$, $M>0$ $(M\ge 0)$ indicates that the matrix $M$ is positive definite (positive semidefinite) and ${\lambda }_{\max }\left(M\right)$ $\left({\lambda }_{\min }\left(M\right)\right)$ denotes its largest (smallest) eigenvalue. The cardinality of a finite set $S$ is denoted as $\left|S\right|$.