## 1 Introduction

Over the past two decades, with the advent of wireless networks and powerful embedded systems, the distributed coordination of multi-agent systems has received significant attention in the control community due to its wide applications in various engineering systems such as data fusion of sensor networks, task cooperation of robots, synchronization of distributed oscillators, and formation maneuver of unmanned vehicles. As the most fundamental research topic for multi-agent coordination, consensus problems have been investigated intensively. Consensus refers to a group of agents reaching an agreement on certain quantities of interest via local interaction. By specifying desired separations among different agents, consensus algorithms can be applied to achieve distributed coordination including formation control and flocking.

The consensus problems have been primarily studied for multi-agent systems with different dynamics (see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and references therein). According to the rate of convergence, which is a significant performance index for evaluating the effectiveness of the designed consensus algorithms, existing consensus studies can be roughly categorized into two classes, namely, asymptotic consensus and finite-time consensus. Asymptotic consensus problems were widely investigated under different scenarios [3, 4, 5, 6, 8, 14, 15, 16]. In [3], under directed switching topologies, asymptotic consensus problems were solved if and only if the time-varying network topologies jointly had a directed spanning tree. Recently, some conditions for second-order consensus were derived in [6, 14, 16]. By using adaptive control approaches, the adaptive consensus problem was studied in [4, 15]. Furthermore, the consensus tracking problem of multiple Euler-Lagrange dynamics was studied in [10, 17].

Different from the asymptotic consensus, achieving consensus in finite time was also studied by many researchers. The finite-time consensus problem was studied in [18] for multiple single-integrator systems, where the signed gradient flows of a differential function and discontinuous algorithms were used. Since then, a variety of finite-time consensus algorithms were proposed to solve the finite-time consensus problem under different scenarios (see [12, 19, 20, 21, 22, 23, 28, 29] and references therein). In [22, 23], the finite-time average consensus problem was investigated for multiple single-integrator systems. Further, a class of finite-time consensus algorithms for multiple double-integrator systems were given in [7, 11, 13, 17, 20, 21]. Then, the finite-time consensus problem for multiple non-identical second-order nonlinear systems was studied in [28]

with the settling time estimation. However, the settling time functions in

[28] depended on initial states of the agents, which prohibited their practical applications if the knowledge of initial conditions was unavailable in advance.Recently, the authors in [24] presented a novel class of nonlinear consensus algorithms under an undirected topology for single-integrator multi-agent networks, called fixed-time consensus which assumed uniform boundedness of a settling time regardless of the initial conditions. The results in [24] were further generalized in [26] to solve the robust fixed-time consensus problems under undirected topologies for single-integrator systems with bounded input disturbances. Due to the nonlinear nature of the fixed-time algorithms, it was very difficult to generalize the existing results for first-order systems [24, 26] to multi-agent systems with more complex agent dynamics. A first attempt was made in [25] for double-integrator systems. Further, in [27], a truly distributed algorithm was given under undirected topology, which depended only on the relative measurements of the neighboring agents. Also, for multiple linear systems, the fixed-time formation problems were studied in [12] under an undirected complete graph. It is worth noting that most of the above-mentioned works were derived for multi-agent systems under undirected topologies. For the case of directed topologies, the existing algorithms in [25] depended directly on the inputs of each agent’s neighbors, which led to a loop problem when there exists cycles in the graph. In practical applications, it is significant and challenging to design truly distributed fixed-time consensus algorithm based only on the relative measurements of the neighboring agents for double-integrator multi-agent systems under directed topologies.

Motivated by the above observations, by using a motion-planning approach, this paper investigates the fixed-time consensus problem of double-integrator systems under directed fixed and switching topologies, respectively. The main results of this paper extend the existing works in three aspects. Firstly, by using a motion-planning approach, a novel framework is introduced to solve the fixed-time consensus problems. In this framework, for double-integrator systems considered in this paper, compared with [24, 25, 26, 27, 12], a class of distributed algorithms are designed under a directed interaction topology, which has a directed spanning tree. Secondly, compared with the existing results in [28], where the finite settling time can only be estimated and related to initial conditions, in this paper, with the proposed fixed-time consensus algorithms, the settling time can be off-line pre-assigned according to task requirements. Unlike the results in [24, 25, 26, 27], the bounded settling time can be off-line designed in advance without estimations. Thirdly, the algorithms designed in this paper are based only on sampling measurements of the relative states among its neighbors, which greatly reduces the cost of the network interaction. To the best of authors’ knowledge, it is the first time to solve the fixed-time consensus problems under directed fixed and switching topologies for double-integrator systems.

## 2 Preliminaries

In this section, we introduce some preliminary knowledge of graph theory and matrix theory for the following analysis.

For a multi-agent system with agents, a directed graph is used to model the interaction among these agents, where is the node set and is the edge set. An edge

is an ordered pair of vertices in

, which means that agent can receive information from agent . If there is an edge from to , is defined as the parent node and is defined as the child node. The neighbors of node are denoted by , and is the cardinality of . A directed tree is a directed graph, where every node, except for the root, has exactly one parent. A directed spanning tree of a directed graph is a directed tree formed by edges that connect all the nodes of the graph. We say that a graph has a directed spanning tree if a subset of the edges forms a directed spanning tree. The interaction topology may be dynamically changing. Therefore let denote the set of all possible directed graphs defined for the agents. In applications, the possible interaction topologies will likely be a subset of . Obviously, has finite elements. The union of a group of directed graphs is a directed graph with nodes given by and edge set given by the union of the edge sets of . The adjacency matrix associated with is defined such that if there is an edge from to , and otherwise. The Laplacian matrix of the graph associated with the adjacency matrix is given as , where and , . Given a matrix , it is said that is nonnegative if all its elements are nonnegative, and is positive if all its elements are positive. Further, if a nonnegative matrix satisfies , where represents with an appropriate dimension, then it is said to be stochastic [37].## 3 Fixed-time consensus under a directed fixed topology

In this section, the fixed-time consensus for multiple double-integrator systems is studied under a directed fixed topology.

Consider the multi-agent system with agents labeled as . The dynamics of each agent is described by

(1) |

where and are, respectively, the position and velocity of agent , and the control input.

###### Definition 1.

The main objective of this section is to design a class of distributed algorithms for multi-agent systems (1) with double-integrator dynamics such that the positions and velocities of all agents in networks reach consensus in a fixed settling time, which can be off-line pre-assigned. To achieve this objective, a motion-planning approach is used to design the following algorithm

where . The time sequence is given by , where , , and is a finite settling time which can be off-line pre-assigned according to task requirements.

###### Remark 1.

It is worth mentioning that the above distributed algorithm (3) is designed based on a motion-planning approach. Concretely, consider the cost function and the associated Hamiltonian function with terminal conditions

where and both represent the co-states. Solve the above optimal planing problem in light of Pontryagin s principle [38]. One obtains the consensus algorithm (3) for multi-agent systems with double-integrator dynamics (1).

###### Assumption 1.

Suppose that the topology among the agents is directed and has a directed spanning tree.

Before moving on, the following lemmas are firstly given.

###### Lemma 1.

, zero is a simple eigenvalue of

withas an eigenvector and all of the nonzero eigenvalues are in the open right half plane.

###### Lemma 2.

[3] Let

be a stochastic matrix. If

has an eigenvalue with algebraic multiplicity equal to one, and all the other eigenvalues satisfy , then is SIA, that is, , where satisfies and . Furthermore, each element of is nonnegative.Then, the following theorem provides the main result in this section.

###### Theorem 1.

Suppose Assumption 1 holds. For an off-line pre-assigned settling time , the distributed algorithm (3) solves the fixed-time consensus problem of the multi-agent system (1) under directed fixed topologies, i.e., , and , when . Further, the final consensus values and are given by

(3) |

where and are the initial states of the agents.

Proof: Firstly, we will prove that the states at time sequence can achieve consensus as . By substituting the consensus algorithm (3) into multi-agent systems (1), the closed-loop system can be obtained as follows:

where . Then, by integrating (3) from to , one has

(5) | |||||

Let and . One has

where . Then, in the matrix form, one has

where

and

Under Assumption 1, the directed fixed topology has a spanning tree. Thus, is a stochastic matrix. According to Lemma 1, one gets that has an eigenvalue with algebraic multiplicity equal to one, and all the other eigenvalues satisfy . Thus, it is followed from Lemma 2 that for matrix

, there exists a column vector

such that(6) |

Besides, according to and , one has . Thus, is bounded. It follows that

(7) |

Denote and . From (1), one has

It follows that

Thus, according to (6) and (7), one has

Thus, one has the discrete states will achieve consensus with an exponential rate as , i.e., .

Secondly, for the off-line pre-assigned settling time , we will prove that the discrete states can achieve fixed-time consensus as .
Since , one has

and

where . Therefore, for the off-line pre-assigned settling time , the discrete states will achieve fixed-time consensus in exponential rate as .

Finally, we will prove that the continuous states can achieve fixed-time consensus as . By integrating equation (3) from to , it is obtained that

(8) | |||||

Thus,

where . Note that with an exponential rate and with a polynomial rate. Thus, one has

Besides, according to and , one has . Further, integrating equation (8) from to , one gets

Therefore,

Thus, one obtains

Note that is upper bounded, and

One has . Based on the above analysis, it follows that with the algorithm (3), the multi-agent systems of double-integrator dynamics (1) can achieve fixed-time consensus. The proof is completed.

###### Remark 2.

Under directed topologies, the finite-time consensus problem of single-integrator multi-agent systems has been solved in [30, 31, 32]. However, the algorithms in [30, 31, 32] are difficult to develop for solving the finite-time consensus problem of double-integrator multi-agent systems under directed topologies. Also, in [24, 26], a fixed-time consensus algorithm is developed for integrator-type multi-agent systems under undirected topologies. In this paper, by using a motion-planning approach, a novel class of distributed algorithms are proposed to solve the finite-time or fixed-time consensus problem of double-integrator multi-agent systems under directed topologies.

###### Remark 3.

Compared with the existing works [21, 19, 11, 28] on finite-time consensus problems and [27, 24, 26, 25] on fixed-time consensus problems, in this paper, the settling time can be off-line pre-assigned according to task requirements, which not only realizes the consensus in the state space but also accurately controls the settling time in the time axis.

## 4 Fixed-time consensus under directed periodically switching topologies

In some cases, the interaction among agents exhibits periodic phenomena, which implies that the topology among agents is periodically time-varying. Thus, we will investigate the fixed-time consensus problems of double-integrator multi-agent systems under directed periodical switching topologies. Before moving on, the following assumption is given.

###### Assumption 2.

For a time series with , there exists a corresponding directed topologies set . The topology among agents is periodically time-varying with the period , (i.e. , and the topologies only exist at the time instant) such that across each time interval , the union of the directed interaction graphs at discrete times has a spanning tree.

Based on Lemma 3, we will analyze the control algorithm (3) under directed periodical switching topologies satisfying Assumption 2. In this case, the notation in (3) is replaced by .

###### Theorem 2.

Proof: Note that if we prove the discrete states will achieve consensus as with an exponential rate, then it follows from Theorem 1 that the conclusion in this theorem can be obtained. Thus, we will prove that discrete states will achieve consensus as with an exponential rate.

Denote For the above defined directed periodical switching topologies satisfying Assumption 2, one has . Therefore, according to Lemma 3, one has

It follows that will convergent to with an exponential rate as . Thus, for multi-agent systems (1), it follows from the proof of Theorem 1 that

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