This paper addresses the problem of consensus tracking with fixed-time convergence, for leader-follower multi-agent systems with double-integrator dynamics, where only a subset of followers has access to the state of the leader. The control scheme is divided into two steps. The first one is dedicated to the estimation of the leader state by each follower in a distributed way and in a fixed-time. Then, based on the estimate of the leader state, each follower computes its control law to track the leader in a fixed-time. In this paper, two control strategies are investigated and compared to solve the two mentioned steps. The first one is an autonomous protocol which ensures a fixed-time convergence for the observer and for the controller parts where the Upper Bound of the Settling-Time (UBST) is set a priory by the user. Then, the previous strategy is redesigned using time-varying gains to obtain a non-autonomous protocol. This enables to obtain less conservative estimates of the UBST while guaranteeing that the time-varying gains remain bounded. Some numerical examples show the effectiveness of the proposed consensus protocols.
In the last years, the problems of coordination and control of Multi-Agent System (MAS) have been widely studied (see for instance [1,2,3,4,5]), due mainly to the ability of a MAS to face complex tasks that a single agent is not able to handle. Distributed control approaches applied to a MAS require a communication network allowing to share information with a subset of agents (neighbors). In this context, several interesting problems and applications have been investigated in the literature, for instance, synchronization of complex networks [6], distributed resource allocation [7], consensus [8] and formation control of multiple agents [9]. Among all the mentioned problems, an interesting one is the leader-follower consensus problem where a set of agents, through local interaction, converge to the state of a leader, even though the leader may not be accessible for all agents.
The consensus problem consists in reaching a common agreement state by exchanging only local information [8,10]. Linear average consensus protocols with asymptotic convergence were proposed in [8,10]. It has been demonstrated that the second smallest eigenvalue of the Laplacian graph (i.e. the algebraic connectivity) determines the convergence rate of the MAS.
Furthermore, the problem of tracking a reference by a MAS (i.e. leader-follower consensus problem) has been investigated where the common agreement to reach is the state of a reference imposed by a leader which evolves independently of the MAS [11,12,13,14]. In [14], the consensus problem has been addressed where the agents reach a time-varying reference. However, the control protocol has been derived for first-order MAS. The problem for second-order MAS has been studied in [12] and extended to high-order MAS in [11,13]. Furthermore, [15] has considered the consensus tracking control problem of uncertain nonlinear MAS with predefined accuracy. Nevertheless, in these works, the convergence is only asymptotic.
To improve the convergence rate of a MAS, finite-time consensus protocols have been investigated in [16]. Finite-time stability has been studied in [17,18,19]. However, the settling time is an unbounded function of the initial conditions of the system. Therefore, the concept of fixedtime stability has been introduced and applied to systems with time constraints [20,21,22]. In this case, the settling time is bounded by a constant which is independent of the initial conditions of the system. In the literature, there are several contributions on algorithms with fixed-time convergence property, such as stabilizing controllers [21,23], state observers [24], multi-agent coordination [25,26], online differentiation algorithms [27,28], etc. Nevertheless, one can mention that the fixed-time stabilization problem of second-order systems is not an easy task since usually the settling time is not provided or is overestimated. Indeed, there are several works for second-order systems stabilization based on block-control techniques ( [29,21,30,31]) or on the homogeneity in the bi-limit ( [32]). However, the homogeneity-based algorithms do not provide an estimate of the settling time and many block-control-based algorithms neglect some transient when the system trajectories stay on a region around a manifold. Moreover, the works [33,34,35,36] deal with the problem of leader-follower consensus. Nevertheless, these algorithms require that each follower know the inputs of its neighbors simultaneously, which causes communication loop problems. In this paper, we address the leader-follower consensus problem of a MAS, where each agent of the MAS estimates and tracks the trajectory of the leader using local available information even when just a subset of MAS has access to the leader state, and we provide the necessary conditions to achieve the convergence in a fixed-time.
A Lyapunov differential inequality for an autonomous system to exhibit fixed-time stability was presented in [21]. Based on this methodology using autonomous systems, the consensus problem with fixed-time convergence property has been derived for first-order MAS in [25,37,38]. Nevertheless, in [38], the UBST has been estimated from design parameters, algebraic connectivity and group order. Thus, it cannot be easily tuned. In [25], the UBST was a design parameter which was established a priory by the user. However, the settling time becomes overestimated and the slack between the settling time and the UBST is conservative. Furthermore, the works [39,40,41] have addressed the consensus tracking problem, i.e., the MAS follows a trajectory imposed by the leader. The scheme presented in [40] has introduced a fixed-time algorithm considering inherent dynamics for the agents. However, disturbances were not taken into account. The leader-follower consensus problem for agents with second-order and high order integrator dynamics has been addressed in [42,41], respectively. The approach was based on a fixed-time observer to estimate the
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