Event based guaranteed cost consensus for distributed multi-agent systems

Highlights

• The concept of guaranteed cost is introduced to incorporate with event based control to further investigate the energy consumption.

• A novel event based consensus approach is applied to general LTI multi-agent systems.

• The controller gain matrix and the triggering parameters with guaranteed cost performance are co-designed to achieve the consensus for distributed multi-agent systems.

• A BMI (bilinear matrix inequality) based approach is used to reduce the conservativeness of controller design.

Abstract

To investigate the energy consumption involved in an event based control scheme, the problem of event based guaranteed cost consensus for distributed multi-agent systems with general linear time invariant dynamics is considered in this paper. A delay system method is used to transform the multi-agent systems into a special delay system based on a sampled-data event triggering mechanism, which only requires supervision of system states at discrete instants. Sufficient conditions to achieve the consensus with guaranteed cost are presented and expressed as a continuous constrained optimization problem with a linear objective function, linear and bilinear matrix inequalities constraints, involving the co-design of the controller gain matrix and event triggering parameters. An illustrative example is given to show the effectiveness of the proposed approach.

Introduction

Multi-agent systems have attracted considerable attention in recent years due to their broad applications in distributed sensor networks, satellite clusters, unmanned aerial vehicle (UAV) formations and robot teams [7], [16], [17], [18], [25], [26]. Distributed consensus is a significant problem in multi-agent systems, in which, a group of agents needs to agree on certain quantities of common interest, only sharing information with their neighbors locally [2], [29], [30]. An important issue in the implementation of distributed consensus algorithms comes from the communication and controller actuation schemes. In traditional time-triggered control schemes, the sensor and controller are updated uniformly with a fixed sampling period regardless whether it is necessary or not. Since each agent may be equipped with an embedded microprocessor with limited computing and communication capabilities, event based control (also called event triggered or event driven control) has emerged as an alternative to time triggered control to reduce the number of actuator updates and to facilitate the efficient usage of shared resources. In an event based control scheme, the controllers are updated only when some specific events occur and therefore the frequency of controller updates is reduced [1].

Previous work on event based consensus of multi-agents systems can be found in [4], [8], [15], [24], [33] and references therein. To sum up, the mode of event detection can be classified into two groups: (1) continuous event detection; (2) sampled-data event detection. In continuous event detection, event generators have to monitor and check the event triggered conditions constantly and should exclude Zeno behavior [4], [8], [24]. Obviously, such continuous detection does not sufficiently meet the original requirements to reduce the communication frequency between control components, which may increase the burden of imbedded microprocessors and become an important source of energy consumption. To address the limitations of continuous detection, the concept of sampled-data event detection was proposed in [15], which admits a minimum inter-event time and it is lower bounded by the sampling period; therefore, the Zeno behavior is prohibited automatically. In this study, we will use the sampled-data event detection.

It should be noted that the majority of existing literatures on event based multi-agent consensus problems focus on the case where agents are governed by first-order or second-order dynamics [4], [8], [15], [24], in which, since the control law is predetermined, there is no necessity to design the control gain matrix. In this study, the event based consensus of multi-agent systems with general linear time invariant (LTI) dynamics is considered. This framework not only admits the wide applicability of multi-agents but also allows us to design the controller appropriately.

In general, a stabilization problem in control design can be formulated as a feasibility problem by Lyapunov stability theory from the perspective of optimization. In some cases, for example, the robustness against uncertainty [11], [13], [20], [27], [28], to guarantee some level of performance, the concept of guaranteed cost control was firstly introduced in [3], which will yield a standard continuous constrained optimization problem. Taking into account the energy consumption, the event based guaranteed cost consensus is studied, with an objective function involving energy consumption added. To the best of our knowledge, this work that emphasizes the necessity of linking the event based control with guaranteed cost performance from the perspective of energy saving is the first time in multi-agent systems.

The main contribution and novelty of this paper can be summarized as follows: (1) the concept of guaranteed cost is introduced to incorporate with event based control to further investigate the energy consumption; (2) a novel event based consensus approach is applied to general LTI multi-agent systems; (3) the controller gain matrix and the triggering parameters with guaranteed cost performance are co-designed to achieve the consensus for distributed multi-agent systems; (4) a BMI (bilinear matrix inequality) based approach is used to reduce the conservativeness of controller design.

Notations: Throughout this paper, the symmetric terms in a symmetric matrix are denoted by $\left(\genfrac{}{}{0em}{}{X}{⁎}\genfrac{}{}{0em}{}{Y}{Z}\right)=\left(\genfrac{}{}{0em}{}{X}{Y^T}\genfrac{}{}{0em}{}{Y}{Z}\right)$; $\mathbf{1}$ and $\mathbf{11}$, denote a vector and a matrix with all ones, respectively; $P>0(P\ge 0)$ means that P is a real symmetric positive definite (semi-definite) matrix; ${\lambda }_{\max }\left(P\right)$ denotes the maximum eigenvalue of matrix P.