Distributed optimal control and $L_2$ gain performance for the multi-agent system with impulsive effects

Abstract

In this paper, we investigate the distributed optimal control and $L_2$ gain performance for the multi-agent system with impulsive effects. First, we obtain sufficient conditions to ensure that the distributed protocols can minimize the desired performance index with state-control cross weighting terms. Second, we propose and solve the bounded $L_2$ gain synchronization problem for the impulsive system with hybrid disturbance inputs. Finally, an example is presented to illustrate the efficiency of the obtained results.

Introduction

Recently people have witnessed an increasing interest in networks of systems [ [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]], especially, the distributed control for multi-agent system due to its wide applications in various areas, such as air traffic control, automated highway systems, and sensor networks [ [11], [12], [13]]. Also, optimality of the protocol is an important topic for multi-agent system (MAS). Discussing the cost function or the performance index aims at getting a balance between the state convergence and the control effort because of the limited resources constraint. Compared with the centralized control, the distributed controller will save more efforts and be more feasible to implement in practice. Ma et al. [14] investigated the optimal consensus of the first-order and the second-order multi-agent systems, respectively, and mentioned that the optimal topology is a star topology under the special performance index. Ma et al. [15] studied the competition phenomena of multi-agent systems consisting of three groups of agents. Hengster-Movric and Lewis [16] and Lewis et al. [17] obtained the optimal distributed consensus protocols by the inverse optimality theory. The main idea of the inverse optimality theory is to analyze the optimality of a stabilizing control with respect to some prescribed performance index [18]. Zhang et al. [19] provided the conditions for globally optimal cooperative control problems of MAS on directed graphs which contain a spanning tree. Liao et al. [20] and Wu et al. [21] investigated the multi-agent systems with error integral and preview action and designed the controller which can achieve the cooperative optimal preview tracking. Xie and Lin [22] studied the global optimal consensus problem for a multi-agent system with bounded controls.

As we know, the actual system is often deeply disturbed by human exploring activities such as planting and harvesting. Impulsive differential equation has a wider application when analyzing many real world phenomena such as the frequency-modulated signal processing systems and flying object motions. Systems with impulse and the potential engineering applications have received extensive attention in the past few years, see Chen et al. [23], Han et al. [24], Wang and Shen [25], Guan et al. [26], Suo and Sun [27], Li and Wu [28] and references therein. However, to the best of our knowledge, to this day, for the MAS with impulse, except for [29], there exist no results which focus on discussing the problem that the distributed control law can make the desired performance, especially with cross weighting terms, reach minimum.

Meanwhile, it is well known that the external disturbance is quite common in real world and we hope that the influence of the external disturbance is as small as possible. In order to find the control law such that the closed-loop networked system reaches the synchronization in absence of disturbances, and that synchronization $L_2$ gain is bounded for all $L_2$ disturbance inputs, it is indispensable to discuss the distributed performance synthesizing problem [ [30], [31], [32], [33]]. For example, Wang et al. [30] considered the consensus problem of MAS with external disturbances using the $H_{\infty }$ control theory. Wang et al. [31] discussed the distributed robust control of uncertain linear multi-agent systems. Amini et al. [32] investigated a new method for consensus in nonlinear multi-agent systems using fixed-order non-fragile dynamic output feedback controller, via an LMI approach. However, to the best of our knowledge, there exists no research on the bounded $L_2$ gain synchronization problem for the impulsive MAS with hybrid disturbance inputs. The study on this issue is meaningful and challenging.

To the best of the authors’ knowledge, this paper is the first work that addresses the problem about the distributed optimal control and $L_2$ gain performance for the multi-agent system with impulsive effects. The detailed contribution is in two aspects. One is to present the distributed protocol to optimize some specified performance index with state-control cross weighting terms, which has wider applications in engineering [34]. The other is to investigate the bounded $L_2$ gain synchronization problem for the impulsive MAS with hybrid disturbance inputs. We transform the hybrid disturbance rejection problem into an optimal control problem, design the distributed control and obtain the worst disturbance case.

This paper is organized as follows. In Section 2, we present the preliminaries. In Section 3, for the impulsive MAS, we discuss the distributed optimal control with state-control cross weighting terms in the performance index. In Section 4, we propose and solve the bounded $L_2$ gain synchronization problem for the impulsive MAS with hybrid disturbance inputs. In Section 5, numerical simulations are presented. The paper ends up with a brief discussion.