Observer-based consensus tracking of multi-agent systems with one-sided Lipschitz nonlinearity

Hongjun Chu, Xiaocheng Liu, Weidong Zhang, Yunze Cai
2016 Journal of the Franklin Institute  
This paper investigates the observer-based consensus tracking problem of multiagent systems with one-sided Lipschitz nonlinearity. The agent dynamics considered here covers a broad family of nonlinear systems, and includes the well-known Lipschitz system as a special case. To achieve consensus tracking for such multi-agent systems, two types of observer-based protocols named the continuous protocol and the intermittent protocol are proposed. Furthermore, several multi-step design algorithms are
more » ... presented to select the observer gains and the controller parameters of the proposed protocols. It is shown that the established sufficient criteria can not only ensure the observer error to approach to zero, but also realize the consensus tracking of multi-agent systems. The obtained results are illustrated by two simulation examples. 2 possesses inherent advantages in the control and observation theory, and in particular to address the observer design problem [9] . However, the existing investigations on the one-sided Lipschitz are only confined in the single system framework. To the best of our knowledge, until now, no results have been found on the study of consensus tracking for multi-agent systems with one-sided Lipschitz, and this motivates our research. The relative states between neighboring agents are not always available, while the relative outputs are easy to acquire. Thus, consensus by output feedback is easier to implement than one by state feedback in practice. Due to static output feedback being quite restrictive [10], observer-based (or dynamic) output feedback is an elegant alternative [11] . Existing results on this topic can be found in the literatures. To track an active leader, a novel distributed estimator-based protocol was designed in [12] for first-order agents. Subsequently, this result was extended to the second-order case in [13] , and the finite-time tracking control in [14] , respectively. For multi-agent systems with general linear dynamics, an observer-based consensus protocol was proposed in [15] by using the relative output information. Besides this full-order observer, the reduced-order observer was further investigated in [16] . A consensus tracking framework, including state feedback and output feedback, was presented for continuous-time multi-agent systems in [17] , and the discrete-time counterpart in [18] . It should be pointed out that most of the existing results with respect to observer-type protocols mainly focus on integrator systems or general linear systems. Until now, few works on nonlinear multi-agent systems are found [19] . The traditional observer problem of a single nonlinear system has been studied for decades [8, 9, [20] [21] [22] , and there are mainly four different approaches to design nonlinear observer [23] . Here, we only consider the case of one-sided Lipschitz agents. More specifically, the systems dynamics can be split into a linear part and a nonlinear part, where the linear part is assumed to be observable from system output, and the nonlinear part is one-sided Lipschitz. On the other hand, the control inputs are inevitably missing in real control systems due to: 1) some unpredictable external factors that break edges or vertexes, e.g., actuator failure, packet loss; 2) some intentional reasons such as communication cost limitation, and the controllers are purposefully suspended from time to time. The occasionally missing control inputs can make control discontinuous, e.g., intermittent control and impulsive control. Compared with the continuous case, the discontinuous control method is more economical and efficient. Therefore, it has aroused great interest among researchers, since it was proposed in [24] . More recently, the intermittent protocols were proposed in [25] to reach consensus tracking of linear multi-agent systems. This intermittent idea was also adopted in containment control of multi-agent systems with intermittent input saturation [26] , and synchronization of complex network via randomly occurring control [27] . However, these results are only restricted in state feedback framework rather than output feedback case. Motivated by the above discussions, this paper aims to design the observer-based protocols for achieving consensus tracking for multi-agent systems with one-sided Lipschitz nonlinearity. Based on relative outputs between agents, we first design distributed state observers, and then, by using the observed states, construct the continuous protocol and the intermittent protocol. In summary, the main contributions of this paper are threefold: (1) we introduce one-sided Lipschitz condition in [9, [20] [21] [22] into multi-agent systems, and then extend the traditional observer design for a single nonlinear system to the observer-based 3 protocols design for multi-agent systems with one-sided Lipschitz; (2) Two types of observer-based protocols, namely the continuous protocol and the periodically intermittent protocol are proposed here. Compared with the former, the latter seems more practical and environment-friendly in engineering. The observer leads to dimension expansion of closed-loop systems, together with nonlinear term and intermittent communication network jointly making convergence analysis intractable. The multiple Lyapunov functions based on dwell time approach is used to overcome this difficulty. Compared to existing results, we extend the state feedback protocols in [2, 3] to the dynamic output feedback protocols, and extend continuous observer-based protocols in [19] to the intermittent case. (3) Multi-step algorithms are given to design the observer gains and the protocol parameters, which make the corresponding protocols easy to implement. Based on these algorithms, the sufficient criteria for achieving the consensus tracking are established. It should be pointed out that these criteria contain the dwell time information regarding topological graphs. Moreover, compared with the existing results, our results are more general and less conservative. The remainder of the paper is organized as follows. Section 2 provides some preliminaries and problem formulation. In Section 3, we construct two kinds of observer-based protocols, and analytically prove that they can solve consensus tracking problem for multi-agent systems with one-sided Lipschitz nonlinearity. Two simulation examples are given in Section 4, and finally, Section 5 concludes the whole paper. H a FR J J J
doi:10.1016/j.jfranklin.2015.10.011 fatcat:kdzfmb354fcydayu6v6pnyvhbq