Output feedback consensus control for fractional-order nonlinear multi-agent systems with directed topologies

Abstract

This paper is devoted to the output feedback consensus control problem for a class of nonlinear fractional-order multi-agent systems (MASs) with general directed topologies. It is worth noting that the considered fractional-order MASs including the second-order MASs as special cases. By introducing a distributed filter for each agent, a control algorithm uses only relative position measurements is proposed to guarantee the global leaderless consensus can be achieved. Also the derived results are further extended to consensus tracking problem with a leader whose input is unknown and bounded. Finally, two simulation examples are provided to verify the performance of the control design.

Introduction

Consensus control of multi-agent systems (MASs) is to design appropriate controller for each agent only using local information between neighbors such that the states of all agents reach general agreement. From the viewpoint of existing number of leaders in MASs, existing consensus problem of MASs can be classified into two categories: leaderless consensus problem [1], [2], [3] and consensus tracking (or leader-follower consensus) problem with a leader [4], [5], [6], [7]. During the last decade, consensus of MASs has received considerable attention and there are many results available in the literature, see the recent survey papers [8], [9] and references therein.

Early work often focuses on consensus of integer-order MASs, i.e., first-order MASs [1], [4], [5], second-order MASs [2], [3], [6], and high-order MASs [7], [10], [11]. However, several phenomena can be explained naturally by the collective group behavior of agents with fractional-order dynamics rather than classical integer-order dynamics [12], [13]. For example, the synchronization motion of multiple agents in fractional circumstances, i.e., viscoelastic materials, macromolecule fluids and porous media. Up to now, some researchers have tackled the consensus control of MASs with fractional-order dynamics. Leaderless consensus problem of nonlinear fractional-order double integrator MASs is studied in [14], [15]. Both leaderless consensus problem and consensus tracking problem are considered in [16] for fractional-order MASs with input time delay. Consensus tracking control problem is also presented for fractional-order single integrator MASs with undirected topology [17] or directed topology [18], [19], and for fractional-order double integrator MASs with undirected topology [20] or directed topology [14], [21]. It should be emphasized that fractional-order systems are an extension to the traditional integer-order ones, which have properties of infinity memory and hereditary due to the existence of a memory term in the model [22].

Note that state feedback-based controllers are mainly based on a restrictive assumption that the state variables of each agent can be measured directly. However, in many real applications, full-state measurements are unavailable due to economical concerns or physical constraints. Especially when there exist multiple states in second-order or higher-order MASs, it is unrealistic to obtain the information of multiple states accurately. Thus, state feedback-based control in those cases should be replaced by output feedback-based control. Some results on output feedback-based consensus control problem are presented in MASs with general linear dynamics [23], [24], with second-order agent dynamics [25], [26] and with high-order agent dynamics [27]. So far, no author has studied the output feedback-based consensus problem for MASs with nonlinear fractional-order dynamics.

In reality, the agents might be affected by the interaction among neighboring agents, but also by its own intrinsic nonlinear dynamic. So the MASs with intrinsic nonlinear dynamics are considered recently in [2], [3], [5], [14], [18]. Since the limited view field or nonuniform sensing ranges of sensors, one agent may be able sense another agent, but not vice versa. The communication topology among the agents, in general is directed. Taking into consideration these practical cases, in this paper, we consider the consensus problem of fractional-order double integrator MASs with intrinsic nonlinear dynamics and general directed topologies using only relative output information. Due to the well-known Leibniz rule for fractional derivatives is invalid [28], how to construct a suitable Lyapunov function for analysing the stability of nonlinear fractional-order MASs is very challenging. The output feedback based consensus control of double integrator MASs in the presence of nonlinear fractional-order dynamics is even more challenging as the communication topology among the agents is not only directed but also local.

The main contributions of this paper are summarized as follows. Firstly, for the leaderless consensus problem of fractional-order MASs with intrinsic nonlinear dynamics and directed graph, a novel distributed algorithm combined with a filter is derived. By generalizing an important nonlinear fractional-order inequality, it is shown that all the agents can achieve global consensus if the interaction graph is strongly connected and the control parameters are chosen properly. Secondly, for the tracking problem with a leader whose input is bounded and unknown to any follower, a newly distributed algorithm combined with a similar filter is further developed to guarantee the tracking error and control input are uniformly ultimately bounded (UUB). Of particular interest is all the algorithms designed in this paper can be implemented only using relative out measurements between neighbors.

The rest of this paper is organized as follows. Some preliminaries are briefly presented in Section 2. Output feedback based leaderless consensus problem is first studied in Section 3. In Section 4, the results are then extended to the output feedback based tracking problem when there exists an unknown leader. Simulation examples and conclusions are outlined in Sections 5 and 6, respectively.

Let ${\mathbb{R}}^{m\times n}$ and ${\mathbb{R}}^m$ be the sets of m × n real matrices and m-dimensional Euclidean space, respectively. Let ${\mathbf{1}}_n\in {\mathbb{R}}^n({\mathbf{0}}_n\in {\mathbb{R}}^n)$ stand for the n × 1 column vector of all ones (zeros) and I_n (O_n) be the n × n identity (zero) matrix. Denote by $\text{diag}(d_1,\dots ,d_n)\in {\mathbb{R}}^{n\times n}$ a diagonal matrix with diagonal entries d₁ to d_n. We use ⊗ to represent the Kronecker product. For a vector $x={[x_1,\dots ,x_n]}^T\in {\mathbb{R}}^n,$ $\parallel x\parallel =\sqrt{x^{T}x}$ denotes the Euclidean norm. Let $\overline{\lambda }\left(A\right)\left(\underline{\lambda }\left(A\right)\right)$ denote the maximal (minimum) eigenvalue of a positive definite matrix $A\in {\mathbb{R}}^{n\times n}$. Let |b| and σ(B) be the absolute value of a real number $b\in \mathbb{R}$ and the maximal singular value of a matrix $B\in {\mathbb{R}}^{m\times n},$ respectively.