Mathematical Problems in Engineering

Volume 2018, Article ID 1957070, 11 pages

https://doi.org/10.1155/2018/1957070

## Predefined-Time Consensus of Nonlinear First-Order Systems Using a Time Base Generator

^{1}Centro de Investigación en Matemáticas (CIMAT), A.C., Computer Science Department, GTO, Mexico^{2}Tecnologico de Monterrey, Escuela de Ingeniería y Ciencias, Zapopan, Mexico^{3}Intel Tecnología de México, Multi-Agent Autonomous Systems Lab, Intel Labs, Zapopan, Mexico

Correspondence should be addressed to Héctor M. Becerra; xm.tamic@arreceb.rotceh

Received 16 June 2018; Revised 11 September 2018; Accepted 17 September 2018; Published 9 October 2018

Academic Editor: Ju H. Park

Copyright © 2018 J. Armando Colunga et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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) [1, 2] and in the synchronization of complex dynamical networks [3–5]. On the one hand, consensus protocols have been applied to flocking [6], formation control [7, 8], and distributed resource allocation [9, 10]. On the other hand, the results on synchronization of complex dynamical networks have been applied to neuroscience [3], power-grids [3], and the chaotic synchronization for secure communication in swarms [11].

Several works have been published proposing consensus and synchronization algorithms for different types of systems, considering static [12–14] and dynamic networks [4, 5]. Regarding first-order agents, the standard protocol (the input of an agent is a linear combination of the errors between the agent’s state and those of its neighbors) achieves consensus if the graph topology is strongly connected [15, 16]. This algorithm achieves consensus also for dynamic topologies, switching among connected graphs [15, 17]. For directed graphs topologies, a common requirement is that the graph contains a spanning-tree (e.g., [18]). All these algorithms are based on linear protocols and achieve consensus in an asymptotic fashion, where convergence rate is a function of the algebraic connectivity of the graph.

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 function have been proposed to achieve consensus to the average value [19, 20], the average-min-max value [21], the median value [22], and the maximum or minimum value [23] of the agent’s initial conditions. Continuous finite-time protocols, based on the function, with , have been introduced in [12, 24]. Some algorithms have been extended to achieve finite-time consensus for strongly connected dynamic topologies [20, 25]. Fixed-time protocols have also been proposed in [26, 27]. In these, there exists a bound for the convergence time that is independent of the initial conditions [28, 29]. The protocols are mainly based on functions of the form , with .

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 [30, 31], linear protocols that use time-varying control gains were proposed for reaching network consensus at a preset time. By using a motion planning strategy, predefined-time consensus protocols based on a time-varying sampling sequence convergent to a specified settling time are proposed in [32, 33]. Preliminary versions of a couple of predefined-time protocols were proposed in [34], taking advantage of time base generators (TBGs), which are parametric time signals that converge to zero in a specified time [35], and can be tracked using feedback controllers.

In this work, a couple of consensus algorithms based on TBGs are proposed as an extension to [34]. One algorithm is based on a linear-feedback, and the other is based on the super-twisting controller for the tracking of the TBG signal. The proposed approach can be applied to systems in which agents belong to the class of first-order controllable linear systems and nonlinear systems that can be linearized by state feedback [36]. Comparisons between the proposed protocols and finite-time, fixed-time, and predefined-time protocols are provided through simulations, showing the advantages of the proposed approach. In contrast with works on predefined-time consensus [30–32], the contribution of this paper is threefold: First, the proposed controllers yield smoother auxiliary control signals (without considering the linearization terms) with smaller magnitudes. Second, there are no parameters in the proposed consensus protocols depending on the algebraic connectivity of the graph considered. Third, a super-twisting controller is presented to deal with perturbed dynamics in the network’s agents. With respect to our preliminary conference version [34], two main extensions are reported herein: (1) the new linear-feedback protocol drives the agents towards the initial conditions average (for undirected graphs) or the weighted sum of the initial conditions (for directed graphs), whereas in our previous work the consensus value was artificially specified; (2) convergence to the consensus value in predefined time is demonstrated even if uncertainty in the knowledge of the initial state is introduced in the controller, at difference of our previous results, where it was assumed that the initial state was perfectly known.

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 [37, 38].

The rest of this paper is organized as follows. Section 2 introduces some background in algebraic graph theory, defines the class of systems for which the proposed method is applicable, and formulates the problem to be solved in this paper. Section 3 presents the proposed consensus protocols: a linear-feedback and a super-twisting controller for the tracking of the TBG signal. Section 4 presents simulation results and a comparison of the proposed controllers with existing approaches. Section 5 provides conclusions.

#### 2. Preliminaries and Problem Definition

##### 2.1. Algebraic Graph Theory

Agents’ communication is represented by a graph , which consists of a set of vertices (nodes or agents in this work) , a set of edges , and a weighted adjacency matrix with nonnegative adjacency elements , i.e., iff . The set of neighbors of agent is denoted by .

*Definition 1 (see [39, 40]). *A graph is called* directed* if there is an entry of that fulfills ; i.e., represents that agent receives the information of the state value of the agent ; otherwise, , i.e., . A graph is called* undirected* if the pairs of nodes are not ordered ; otherwise, , i.e., .

*Definition 2 (see [39, 40]). *A directed path (respectively, undirected path) from nodes to is a sequence of distinct edges of the form in a directed graph (respectively, undirected graph). A graph is strongly connected (respectively, connected) if there exists a directed path (respectively, undirected path) between any two distinct vertices and in .

*Definition 3 (see [16, 18]). *A directed graph has a directed spanning tree if there exists a node (a root) such that all other nodes can be linked to via a directed path.

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 [16, 18].

*Definition 4 (see [16]). *Let be a graph on vertices labelled as . The* Laplacian matrix* of is defined as the matrix , where

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 [2, 16, 18].

Lemma 5 ([2, 16, 18]). * has at most one zero eigenvalue with as the corresponding right eigenvector, where denotes a column vector with all entries equal to one, i.e., , and . Furthermore, if the graph is undirected and connected, has a left eigenvector that satisfies and , where represents a column vector of zeros. If the graph is directed and has a spanning tree then and .*

Finally, let us introduce a matrix transformation that will be useful later on. Let be the Jordan form associated with . Then, there exists a nonsingular matrix such that where is the Jordan matrix associated with the nonzero eigenvalues of [39, 41] and represents a column vector of zeros of size .

##### 2.2. Problem Definition

In this paper, we will consider a set of agents connected through a network, which are modeled as first-order nonlinear systems as followsWhere, for the agent is the system’s state, is the control input, and are smooth nonlinear functions, and is a bounded disturbance. In order to represent the multiagent system (MAS) dynamics, denote , , , , and . Then, the complete dynamics (3) can be written as

Let us propose a linearizing control law of the form for each agent (3), where is an auxiliary control input and it is assumed that , i.e., the relative degree is well defined [36] and the agent is controllable. The closed-loop dynamics for the -th agent are now linear and given by

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

In a MAS, the consensus error of agent with respect to its neighbors is defined as [2]

The consensus error for the complete MAS can be expressed in a compact form as [2]**Problem statement**: given a MAS with agents modeled as first-order nonlinear systems (3) and with an associated graph , design a consensus protocol for each agent in the form , such that the state of all the agents reaches a consensus value in a* predefined time* from any initial state , i.e., , as . Therefore, the consensus error (8) converges to zero at time .

*Remark 6. *The consensus problem in MASs is closely related to the synchronization problem in complex dynamical networks, but they have been addressed from different viewpoints [42]. In fact, the consensus problem can be posed as a complete-synchronization problem. In the former, the analysis is often based on the matrix Laplacian approach [2] and the agents have simple dynamics such as single integrator dynamics, a chain of integrators, or linear systems. The recent focus has been on defining finite-time, fixed-time, and predefined-time protocols for fixed and dynamic networks. In the synchronization problem in complex networks the analysis is often based in the Master stability function formalism [3, 42] and addresses interconnected systems with complex dynamics, for instance, chaotic attractors. Although recently some works have addressed synchronization in dynamic networks, e.g., Markov jump topologies [4, 5], most of the work has been focused on static topologies. For further details on the synchronization problem, we refer the reader to [3].

##### 2.3. Time Base Generator (TBG)

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 [35]. For the case of single first-order systems, TBGs have been used for the control of robots at kinematic level [43].

A* time base generator (TBG)* is a continuous and differentiable time-dependent polynomial function described as [35]where is the time basis vector and is a vector of coefficients of proper dimensions. For first-order systems, it is required to compute a TBG fulfilling the following conditions at initial time and final time ,

In [35], a couple of procedures for calculating the coefficients , fulfilling the required constraints, have been introduced. In particular, when a high degree polynomial is used, an optimization procedure can be applied to find the best coefficients.

#### 3. Consensus Protocols Using the TBG

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 , while in Section 3.3 we consider perturbed systems.

##### 3.1. Previous Results

In [34], a consensus protocol was proposed for the MAS (4) by introducing a time-varying control gain in order to drive the system to a neighborhood about a given consensus value in a predefined time , from any initial state . Consequently, the consensus error (8) converges to a neighborhood about the origin at . Such consensus protocol is similar in nature to those in [30, 31].

In the same preliminary work [34], we have proposed the following open-loop controller to enforce predefined-time convergence to a desired consensus value , provided that the MAS (4) is not perturbed and the initial state is known, where fulfills (10). In that work, it was demonstrated that consensus to is achieved in time and the resulting consensus error trajectory in the interval is given by where is the consensus error vector (8) at the initial condition .

To provide closed-loop stability of the tracking error, a different protocol was proposed in [34] to address the predefined-time consensus problem as a trajectory tracking problem, where the reference trajectory depends on the TBG. Such protocol is described as follows,

This protocol guarantees stability during the transient behavior and also after the settling time, i.e., for any . The controller for each agent in (13) consists of two terms: the first one is a feed-forward term that yields the evolution of each agent’s error from the initial value to the origin at time , following a transient profile defined by the TBG as ; the second term is a feedback term that corrects any deviation of from the transient profile , providing global stability of the error dynamics.

##### 3.2. Consensus Based on the Error Feedback

In the conference paper [34], whose main results were summarized above, consensus protocols were proposed achieving predefined-time convergence to a consensus value , which must be specified in the control law. Moreover, it was assumed that the initial state is accurately known since this value is used in the protocols.

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 [15]. The consensus value may represent important information to be recovered, as in a network of sensors. For this reason, in this paper, we propose a new consensus protocol that yields predefined-time convergence of the unperturbed MAS (4) without the need of specifying a consensus value. Moreover, we consider uncertainties in the knowledge of the initial conditions.

In the sequel, consider that the consensus protocol has an estimate of the initial state, denoted as , instead of the real initial state . In this way, where is a vector of uncertainty in the initial state. From (14), the uncertain initial consensus error can be computed 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 introduced in the controller.

Theorem 7. *Consider a MAS with a connected undirected communication topology or directed topology having a spanning tree modeled as in (4). Let be a constant state-feedback gain. Then, the time-varying feedback control law with as in (10) and computed from the initial state estimate , provides global asymptotic stability of the tracking error . Therefore, assuming , predefined-time convergence of the MAS (4) to a consensus value is achieved at from any initial state .*

*Proof. *First, let us demonstrate the stability of the tracking error. The dynamics (3) with , under the control action (16), can be written in vector notation as Computing from (8) and using the linearized dynamics (17), we haveBy using , the initial tracking error is computed as with , where is the Jordan form associated with and is a nonsingular matrix [39].

Let , then , and according to (2), it follows that the first entry of is null, i.e., . Now, taking the time derivative of and using (18), we have Let , and , where and . Then, we can verify that (20) can be written as where and is the Jordan matrix associated with for , .

It can be shown that the solutions of the differential equations (21) are the following [44]: where is the exponential matrix. Then, it follows that since the integral is a finite value due to , and has eigenvalues with negative real parts [18, 45]. Notice that the convergence speed is determined by the dominant eigenvalue of ([15]) and the control gain .

Since , then Therefore, the tracking error converges to the origin. In fact, from (21) it can be seen that the tracking error is globally asymptotically stable, since the real part of the eigenvalues of is negative, , and can be seen as a disturbance that becomes null for . The stability of the tracking error implies that follows ; thus converges to the origin at the predefined-time . Any small final error in time is corrected by the control input , which is applied to each agent for .

By using the property (in accordance to Lemma 5), the converge of the consensus error (8) to the origin implies i.e.,That is, the system (4) achieves predefined-time convergence to a consensus value at using the consensus protocol (16).

Now, let us compute the final consensus value. First, define , where is defined in Lemma 5. For the two types of considered graphs, is an invariant quantity, sincewhere by Lemma 5. Thus, Moreover, from (25) it follows that Then, for a connected undirected graph and by Lemma 5 (), it is obtained that and solving for , one has Hence, the consensus value is the average of the initial state for connected undirected graphs. Now, for a directed graph with a directed spanning tree and by Lemma 5 (), it results in Therefore, the consensus value is given by (32) for a directed graph with a directed spanning tree.

Thus, system (4) with achieves predefined-time convergence to a consensus value at in the presence of uncertainty in the initial state using the consensus protocol (16).

*Remark 8. *It is worth noting that, in contrast to the use of the consensus protocol (13), the feed-forward term of in (16) is not sufficient to drive the consensus error trajectory to the TBG reference trajectory even for ; consequently, system (4) under the first term of (16) does not reach predefined-time convergence to a consensus value at . In such a case, the final state is given by , where denotes the identity matrix of size . Therefore, the use of the feedback term in (16) is required in any case.

It can be seen that the consensus protocol (16) can solve the problem of predefined-time consensus stated in Section 2.2 by tracking the trajectories given by the TBG even when considering uncertainty in the initial state. In this sense, the compensation term for tracking must guarantee reaching the TBG trajectories before the desired convergence time . This means that the tracking control gain must be higher for a larger uncertainty in the initial state. Then, there is a limit on the values of that can be managed by the consensus protocol.

*Remark 9. *Theorem 7 represents a novel contribution on consensus for nonlinear MAS: First, the predefined-time consensus is solved for nonlinear first-order systems. Second, the auxiliary control inputs in (16) are smooth signals even at , which is not the case of other approaches reported in the literature. Third, the proposed consensus protocols do not have parameters depending on the algebraic connectivity of the graph considered. Finally, the stability analysis is performed by using standard techniques for linear systems (upon feedback linearization of the original systems), while most of the works regarding finite-time and fixed-time and even the few existing works on predefined-time convergence adopt more sophisticated techniques such as homogeneity theory and Lyapunov functions.

*Remark 10. *The result introduced by Theorem 7, compared to our previous conference work [34], is twofold: (1) a new consensus protocol is proposed, achieving predefined-time convergence of the unperturbed MAS (4) without the need of specifying a consensus value; (2) the effect of considering uncertainty in the knowledge of the initial conditions is analyzed, demonstrating that convergence to a consensus value can still be achieved.

##### 3.3. Robust Predefined-Time Consensus

In the previous consensus protocol, we have considered unperturbed dynamics (i.e., ). To effectively deal with disturbances, a robust protocol is introduced in this section, by combining the super-twisting controller (STC), which can compensate for matched uncertainties/disturbances [46], with the TBG, leading to a robust predefined-time controller.

In this scheme, it is assumed that there exits a leader in the network system, whose dynamics can be modeled as in [47] by where is the leader’s control input. In this case, agents are now the followers with dynamics (3). The consensus error for each follower is defined as [47]where represents the leader’s adjacency with if agent is a neighbor of the leader; otherwise .

The tracking error is given by where is the TBG reference trajectory for the -th and is computed from the initial state estimate as in (15). The tracking error will be used as the sliding surface; thus

Theorem 11. *Consider a MAS modeled as in (4) with a leader (33). For each agent , define and the sliding surface (36). Consider a function as in (10). There exist gains and , for some function and a constant , such that the following nonlinear controller defined for each agent achieves predefined-time convergence to the leader’s state at , allowing , from any initial state .*

*Proof. *First, taking the time derivative of the tracking error (35) and using the dynamics of the followers (3) and the leader (33), it is obtained thatNow, using (38) and plugging the control law (37) in the time derivative of the sliding surface (36), it results in where . Let ; then the previous expression can be rewritten as It has been proven in [46] that, for a bounded continuously differentiable disturbance, i.e., if and for some constants , , the second-order dynamics (40) converge globally to the origin () in finite time in spite of the disturbance, if adequate positive control gains and are used. Moreover, the remaining dynamics of the tracking error system are constrained to the sliding surface, such that , meaning that each agent follows . Consequently, the consensus error converges to zero in the predefined-time and the consensus value is given by the leader’s state [47].

*Remark 12. *To the best of our knowledge, the only works that propose predefined-time consensus protocols by other authors are [30–32]. The consensus control law (37) has the advantage, over those protocols, of providing robustness against large matched perturbations.

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 , considering that a small will result in large control inputs. Thus the maximum allowable input of each agent must be taken into consideration to set . Third, we are currently assuming no time delay in the agent’s communication; however, time delays may exist in some systems and may affect the convergence to the consensus.

#### 4. Simulation Results

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 (3) with for agents 1 to 4, for agents 5 to 8, and for all the 8 agents. The convergence time is preset to seconds. For this, the TBG is described by the function , which fulfills (9), i.e., , , and . Both proposed distributed control laws are implemented as defined in (16) and (37). The implementation essentially requires the computation of the TGB reference and its time derivative, which means to evaluate the functions and at each iteration and compute the consensus error for each agent depending on the neighbors state as given by (7).

We use the communication topologies shown in Figure 1 (obtained from [30]), denoted as and , where is a connected undirected graph and is a directed graph having a directed spanning tree. The simulations were implemented in MATLAB using the Euler forward method to approximate the time derivatives with a time step of 0.1 ms.