The tracking control of multiagent dynamical systems with exogenous disturbances is studied. A path following algorithm with a time-varying reference state is proposed, and the path tracking of multiagent systems with exogenous disturbance is analyzed. Under the influence of the disturbances, a disturbance observer is developed to estimate the exogenous disturbances. Asymptotical consensus of the multiagent systems with time-varying reference state and exogenous disturbances under the disturbance observers-based control (DOBC) can be achieved for fixed and switching topologies. Finally, by applying an example of multiagent systems with switching topologies and exogenous disturbances, the consensus tracking of multiagent systems with time-varying reference state is reached under the DOBC with the designed parameters.

Recently, the distributed cooperation problems of multiagent systems have attracted many researchers from a broad range of disciplines including physics, biology, computer science, and control engineering. This is partly because of extensive application in many areas such as collaborative control of mobile robots, unmanned air vehicles (UAVs), autonomous underwater vehicles (AUVs), and automated highway systems [

Collective behaviors of autonomous individuals have been extensively studied from different application fields. Consensus that a team of agents gradually reaches an agreement on a common reference value by negotiating with their neighbours has attracted more and more attention recently. Based on the neighbour-based swarm model proposed by Vicsek et al. [

In the current studies, many variables of the leader in multiagent systems are researched. Distributed estimation via observers for multiagent cooperation has been an important topic in the study of agent networks. Peng and Yang studied a leader-following problem for multiagent systems with a time-varying leader and time-varying delays [

In this paper, we investigate the path tracking of multiagent systems with time-varying reference state and exogenous disturbances. By viewing the time-varying reference state as virtual leader, all agents eventually move ahead at a desired tracking under a disturbance observer-based control (DOBC). Disturbance observers (DOs) for dynamic systems have been studied and applied in engineering over two decades. With the development of DOs for robot control independently, DOs have been used in many mechanistic dynamic systems including disk drivers, machining centers, dc/ac motors, and manipulators [

In this paper, we apply the method of DOBC to stabilize the states of multiagent systems with time-varying reference state and exogenous disturbances; this is the novelty of the paper. The rest of the paper is organized as follows. In Section

A weighted connected graph

The set of neighbors of node

A directed path is a sequence of edges in a directed graph

A dynamical equation of multiagent systems was considered in [

Multiagent systems composed of

The moving state of system (

Suppose that we want to stabilize multiagent systems (

In this paper, we discuss the path following of the multiagent dynamical systems with disturbances. The objective is to design a DOBC technique such that the expected reference path (

In this subsection, we suppose there is no disturbance or the disturbance is measurable. The tracking consensus of system (

Here, the controlled system (

The consensus tracking control is given as

Assume the directed graph of

Suppose the polynomial

Substituting

Then, the polynomial

Assume that multiagent systems are composed of

Let

Let

Equation (

The characteristic equation can be rewritten as

The polynomial

Based on Lemma

Suppose the topology of multiagent systems is symmetrical; the multiagent systems (

Since the characteristic value

It is supposed that the states of the system (

Then, we can get that the disturbance observers (

Consider system (

Since

Consider multiagent systems (

Because of the disturbances in the system, the control

Substituting (

Let

Equation (

Suppose the control gain of tracking controls (

Consider system (

Consider multiagent systems (

The proof is similar to that of Theorem

and

If the gain matrix

Consider directed connected networks with

Switching graph of multiagent systems.

Suppose that disturbances in multiagent systems do not exist (or there exist measurable disturbances). Let the initial states of agents be generated randomly in

States tracking of multiagent systems without disturbance.

Consider the multiagent systems with disturbances. Suppose that the parameters of the exogenous disturbance system (

By applying linear matrix inequalities (LMIs) packages of Matlab solving inequality (

States of multiagent systems with disturbance.

Disturbances estimated by DOBC.

States of multiagent systems with DOBC.

The problem of cooperation in a team of unmanned systems with the expected path tracking is considered under the influences of disturbances. A control strategy is designed for the multiagent systems without disturbances, where multiple agents’ states can be controlled to a reference path. Under the effects of the disturbances, disturbance observers are developed by an exogenous system to estimate the disturbances. The disturbance observers are integrated with the controller by replacing the disturbance in the control law with its estimation obtained from the disturbance observer. Furthermore, asymptotical consensus of the multiagent systems with disturbances under the composite controller is achieved.

The authors declare there is no conflict of interests regarding the publication of this paper.

This research is supported by the National Basic Research Science Program of China (under Grant 2012CB720000, 973 Program), the National Natural Science Foundation of China (under Grants 61273152, 61170161, 61174085, and 61304052), the Natural Science Foundation of Shandong Province of China (under Grant ZR2011FM017), and the Outstanding Young Research Award Fund of Shang Province of China (under Grant BS2013DX005).