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This paper proposes a couple of consensus algorithms for multiagent systems in which agents have first-order nonlinear dynamics, reaching the consensus state at a predefined time independently of the initial conditions. The proposed consensus protocols are based on the so-called time base generators (TBGs), which are time-dependent functions used to build time-varying control laws. Predefined-time convergence to the consensus is proved for connected undirected communication topologies and directed topologies having a spanning tree. Furthermore, one of the proposed protocols is based on the super-twisting controller, providing robustness against disturbances while maintaining the predefined-time convergence property. The performance of the proposed methods is illustrated in simulations, and it is compared with finite-time, fixed-time, and predefined-time consensus protocols. It is shown that the proposed TBG protocols represent an advantage not only in the possibility to define a settling time but also in providing smoother and smaller control actions than existing finite-time, fixed-time, and predefined-time consensus.

Recent years have seen an increasing interest in algorithms allowing a group of systems to reach a common value for its internal state through local interaction. This problem has been addressed, from different viewpoints, in the consensus of multiagent systems (MASs) [

Several works have been published proposing consensus and synchronization algorithms for different types of systems, considering static [

With the aim of developing consensus protocols satisfying real-time constraints, finite-time consensus has received a great deal of attention. The main focus has been in defining nonlinear protocols evaluated either on each of the neighbors’ errors or on the sum of the neighbors’ errors. For finite-time convergence, binary protocols based on the

The methods mentioned above cannot be used in applications where real-time constraints have to be accomplished, since in both finite-time and fixed-time consensus, the relationship between the protocol parameters and the convergence time is not straightforward. Thus it is difficult to define the convergence time bound as a function of the protocol parameters. Moreover, the convergence time bound is often overestimated.

To address the consensus design problem with real-time constraints, predefined-time convergence has to be investigated. In this, the time at which consensus is achieved is predefined a priori as a parameter of the consensus protocol, being independent of the initial conditions. Few works have addressed the predefined-time consensus problem. In [

In this work, a couple of consensus algorithms based on TBGs are proposed as an extension to [

It is known that second-order systems can represent a broader class of real systems than first-order systems. However, the last class of systems has also broad applicability, for instance, in robotics for kinematic control of holonomic robots. We are very interested in generalizing our results even for systems of a higher order than two, taking into account existing results like the ones in [

The rest of this paper is organized as follows. Section

Agents’ communication is represented by a graph

A graph

A directed path (respectively, undirected path) from nodes

A directed graph

If an undirected graph has a directed spanning tree, then the graph is connected. A strongly connected directed graph contains at least one directed spanning tree [

Let

In this work, we will propose and analyze consensus protocols for two kinds of networks: connected undirected graphs and directed graphs with a directed spanning tree [

Finally, let us introduce a matrix transformation that will be useful later on. Let

In this paper, we will consider a set of

Let us propose a linearizing control law of the form

The complete closed-loop dynamics for all the agents can be written as

In a MAS, the consensus error of agent

The consensus error for the complete MAS can be expressed in a compact form as [

The consensus problem in MASs is closely related to the synchronization problem in complex dynamical networks, but they have been addressed from different viewpoints [

Time base generators are parametric functions of time, particularly designed to stabilize a system in such a way that its state describes a convenient transient profile. TBGs have been previously used to achieve predefined-time convergence of first and higher order dynamics for single systems in [

A

In [

In this section, we present consensus protocols that take advantage of the TBG to enforce convergence of a MAS in predefined time. In the first two subsections, we present results for unperturbed systems, i.e., for

In [

In the same preliminary work [

To provide closed-loop stability of the tracking error, a different protocol was proposed in [

This protocol guarantees stability during the transient behavior and also after the settling time, i.e., for any

In the conference paper [

Nevertheless, in most of the protocols proposed in the literature, the consensus value is a quantity that inherently results from the initial states and the connectivity of the graph model [

In the sequel, consider that the consensus protocol has an estimate of the initial state, denoted as

The following theorem introduces one of the main results of this paper, a feedback-based consensus protocol able to achieve consensus without the need of specifying the consensus value and able to track the trajectories given by the TBG when there is uncertainty in the initial state

Consider a MAS with a connected undirected communication topology or directed topology having a spanning tree modeled as in (

First, let us demonstrate the stability of the tracking error. The dynamics (

Computing

By using

Let

Let

It can be shown that the solutions of the differential equations (

Since

Therefore, the tracking error converges to the origin. In fact, from (

By using the property

That is, the system (

Now, let us compute the final consensus value. First, define

Moreover, from (

Then, for a connected undirected graph and by Lemma

Hence, the consensus value

Therefore, the consensus value is given by (

Thus, system (

It is worth noting that, in contrast to the use of the consensus protocol (

It can be seen that the consensus protocol (

Theorem

The result introduced by Theorem

In the previous consensus protocol, we have considered unperturbed dynamics (i.e.,

In this scheme, it is assumed that there exits a leader

The tracking error is given by

Consider a MAS modeled as in (

First, taking the time derivative of the tracking error (

Now, using (

It has been proven in [

To the best of our knowledge, the only works that propose predefined-time consensus protocols by other authors are [

Some important issues to be considered during real-world applications of the proposed distributed control protocols are threefold: First, all the clocks of the agents in the network must be synchronized to achieve predefined-time convergence. Second, physical constraints of the systems must be taken into account to set

In this section, simulation results are shown to illustrate the advantages of the proposed protocols. All the simulations consider a MAS of 8 agents, described by (

We use the communication topologies shown in Figure

Communication graphs. Left: undirected graph

The initial states of the eight agents are

This subsection is devoted to showing the performance of the TBG protocol (

Predefined-time TBG-tracking controller (

Predefined-time TBG-tracking controller (

It can be seen in Figure

In this subsection, first, we compare the proposed control law (

Preset-time controller [

Prescribed-time controller [

Specified-time controller [

Additionally, we can see that the specified-time consensus protocol in Figure

Following the comparison, we evaluate the TBG controller (

The results are shown in Figures

Comparison of the TBG-tracking controller (

Comparison of the TBG-tracking controller (

All the previous simulations considered ideal dynamics in (

Predefined-time TBG-tracking controller (

In order to deal with the described matching perturbations

Robust predefined-time TBG controller (

Robust predefined-time TBG controller (

In this paper, a couple of predefined-time consensus protocols for first-order nonlinear MAS under both undirected and directed communication topologies have been proposed. In these protocols, the convergence time to achieve consensus can be set by the user, and it is independent of the initial state conditions. For this, the TBG (time base generator) is combined with feedback controllers to achieve closed-loop stability and robustness. In particular, one protocol uses the super-twisting controller to deal with perturbations. The performance of the proposed controllers has been compared with existing finite-time, fixed-time, and predefined-time controllers through simulations. The results have shown that the proposed protocols achieve consensus in the predefined time, independently of the initial conditions, and exhibit closed-loop stability. Moreover, a benefit of the proposed controllers is that they yield smoother control signals with smaller magnitude than the existing approaches reported in the literature. Furthermore, the proposed schemes can reach predefined-time consensus even when there exists uncertainty in the knowledge of the initial conditions, and the super-twisting protocol is robust against matching perturbations.

Future works will focus on extending the results presented in this paper to the case of high-order nonlinear MAS, as well as in considering the predefined-time consensus problem in Markovian jump topologies [

The paper is a theoretical result on consensus with predefined-time convergence. All proofs are presented, and simulations are provided to illustrate the result. In the authors’ opinion, there is no need to provide the numerical experiments; simulations can be provided and made available under request.

The authors declare that there are no conflicts of interest regarding the publication of this manuscript.

The first two authors were supported in part by Conacyt [grant No. 220796]. The authors would also like to acknowledge the partial support of Intel Corporation for the development of this work.

_{∞}synchronization for complex networks with semi-Markov jump topology