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

Authors: J. Armando Colunga; Carlos R. Vázquez; Héctor M. Becerra; David Gómez-Gutiérrez
Journal: Mathematical Problems in Engineering
 (2018)

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## 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.


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## Body

## 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 thesign(•) 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 •αsign(•) function, with 0<α<1, 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 •psign(•)+•qsign(•), with 0<q<1<p.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. Section2 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 graphG=(V,E,A), which consists of a set of vertices (nodes or agents in this work) V=1,…,N, a set of edges E⊆V×V, and a weighted adjacency matrix A=[aij] with nonnegative adjacency elements aij, i.e., aij>0 iff (i,j)∈E. The set of neighbors of agent i is denoted by Ni=j∈V:j,i∈E.Definition 1 (see [39, 40]).
A graphG is calleddirected if there is an entry of A that fulfills aij>0; i.e., (i,j)∈E represents that agent j receives the information of the state value xi of the agent i; otherwise, aij=0, i.e., (j,i)∉E. A graph G is calledundirected if the pairs of nodes are not ordered aij=aji>0; otherwise, aij=aji=0, i.e., (i,j),(j,i)∉E.Definition 2 (see [39, 40]).
A directed path (respectively, undirected path) from nodesvj to vi is a sequence of distinct edges of the form (vi,vi1),(vi1,vi2),…,(vil,vj) 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 vi and vj in G.Definition 3 (see [16, 18]).
A directed graphG has a directed spanning tree if there exists a node vi (a root) such that all other nodes can be linked to vi 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]).
LetG be a graph on N vertices labelled as 1,…,N. TheLaplacian matrix of G is defined as the N×N matrix L=[lij], where (1)lij=-aij,ifi≠j,∑k=1,k≠iNaik,ifi=j.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]).
L has at most one zero eigenvalue λ1(L) with 1N as the corresponding right eigenvector, where 1N denotes a column vector with all entries equal to one, i.e., λ1(L)=0, Re⁡(λi(L))>0,∀i=2,…,N and L1N=0N. Furthermore, if the graph is undirected and connected, L has a left eigenvector γ that satisfies γTL=0NT and γ=1N, where 0N represents a column vector of zeros. If the graph is directed and has a spanning tree then γTL=0NT and γT1N=1.Finally, let us introduce a matrix transformation that will be useful later on. LetJ∈CN×N be the Jordan form associated with L. Then, there exists a nonsingular matrix S∈CN×N such that (2)S-1LS=J=00N-1T0N-1J^where J^∈C(N-1)×(N-1) is the Jordan matrix associated with the nonzero eigenvalues of L [39, 41] and 0N-1 represents a column vector of zeros of size N-1.


### 2.2. Problem Definition

In this paper, we will consider a set ofN agents connected through a network, which are modeled as first-order nonlinear systems as follows(3)x˙it=fixi+gixiuit+ρit,i∈1,…,NWhere, for the agent i,xi(t)∈R is the system’s state, ui(t)∈R is the control input, fi(xi) and gi(xi) are smooth nonlinear functions, and ρi(t)∈R is a bounded disturbance. In order to represent the multiagent system (MAS) dynamics, denote x=[x1,…,xN]T, u=[u1,…,uN]T, f=[f1,…,fN]T, g=[g1,…,gN]T, and ρ=[ρ1,…,ρN]T∈RN. Then, the complete dynamics (3) can be written as (4)x˙t=fx+gxut+ρt.Let us propose a linearizing control law of the formui=(-fi(xi)+vi)/gi(xi) for each agent (3), where vi is an auxiliary control input and it is assumed that gi(xi)≠0, i.e., the relative degree is well defined [36] and the agent is controllable. The closed-loop dynamics for the i-th agent are now linear and given by(5)x˙it=vit+ρit,i∈1,…,N.The complete closed-loop dynamics for all the agents can be written as(6)x˙t=vt+ρtwhere v=[v1,…,vN]T∈RN.In a MAS, the consensus error of agenti with respect to its neighbors is defined as [2](7)eit=∑j∈Niaijxjt-xit,i∈1,…,N.The consensus error for the complete MAS can be expressed in a compact form as [2](8)et=e1t,…,eNtT=-Lxt.Problem statement: given a MAS with agents modeled as first-order nonlinear systems (3) and with an associated graph G, design a consensus protocol for each agent in the form ui=γ(ei,xi,t), such that the state of all the agents reaches a consensus value x∗ in apredefined time  tf from any initial state x(0), i.e., xi(t)→x∗,∀i={1,…,N}, as t→tf. Therefore, the consensus error (8) converges to zero at time tf.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].Atime base generator (TBG) is a continuous and differentiable time-dependent polynomial function h(t) described as [35](9)ht=τt·Cift∈0,tf0otherwisewhere τ(t)=[trtr-1…t1] is the time basis vector and C 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 tf, (10)ht=1,ift=00,ift≥tfh˙t=0,ift=0ort≥tf.In [35], a couple of procedures for calculating the coefficients C, 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.


## 2.1. Algebraic Graph Theory

Agents’ communication is represented by a graphG=(V,E,A), which consists of a set of vertices (nodes or agents in this work) V=1,…,N, a set of edges E⊆V×V, and a weighted adjacency matrix A=[aij] with nonnegative adjacency elements aij, i.e., aij>0 iff (i,j)∈E. The set of neighbors of agent i is denoted by Ni=j∈V:j,i∈E.Definition 1 (see [39, 40]).
A graphG is calleddirected if there is an entry of A that fulfills aij>0; i.e., (i,j)∈E represents that agent j receives the information of the state value xi of the agent i; otherwise, aij=0, i.e., (j,i)∉E. A graph G is calledundirected if the pairs of nodes are not ordered aij=aji>0; otherwise, aij=aji=0, i.e., (i,j),(j,i)∉E.Definition 2 (see [39, 40]).
A directed path (respectively, undirected path) from nodesvj to vi is a sequence of distinct edges of the form (vi,vi1),(vi1,vi2),…,(vil,vj) 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 vi and vj in G.Definition 3 (see [16, 18]).
A directed graphG has a directed spanning tree if there exists a node vi (a root) such that all other nodes can be linked to vi 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]).
LetG be a graph on N vertices labelled as 1,…,N. TheLaplacian matrix of G is defined as the N×N matrix L=[lij], where (1)lij=-aij,ifi≠j,∑k=1,k≠iNaik,ifi=j.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]).
L has at most one zero eigenvalue λ1(L) with 1N as the corresponding right eigenvector, where 1N denotes a column vector with all entries equal to one, i.e., λ1(L)=0, Re⁡(λi(L))>0,∀i=2,…,N and L1N=0N. Furthermore, if the graph is undirected and connected, L has a left eigenvector γ that satisfies γTL=0NT and γ=1N, where 0N represents a column vector of zeros. If the graph is directed and has a spanning tree then γTL=0NT and γT1N=1.Finally, let us introduce a matrix transformation that will be useful later on. LetJ∈CN×N be the Jordan form associated with L. Then, there exists a nonsingular matrix S∈CN×N such that (2)S-1LS=J=00N-1T0N-1J^where J^∈C(N-1)×(N-1) is the Jordan matrix associated with the nonzero eigenvalues of L [39, 41] and 0N-1 represents a column vector of zeros of size N-1.


## 2.2. Problem Definition

In this paper, we will consider a set ofN agents connected through a network, which are modeled as first-order nonlinear systems as follows(3)x˙it=fixi+gixiuit+ρit,i∈1,…,NWhere, for the agent i,xi(t)∈R is the system’s state, ui(t)∈R is the control input, fi(xi) and gi(xi) are smooth nonlinear functions, and ρi(t)∈R is a bounded disturbance. In order to represent the multiagent system (MAS) dynamics, denote x=[x1,…,xN]T, u=[u1,…,uN]T, f=[f1,…,fN]T, g=[g1,…,gN]T, and ρ=[ρ1,…,ρN]T∈RN. Then, the complete dynamics (3) can be written as (4)x˙t=fx+gxut+ρt.Let us propose a linearizing control law of the formui=(-fi(xi)+vi)/gi(xi) for each agent (3), where vi is an auxiliary control input and it is assumed that gi(xi)≠0, i.e., the relative degree is well defined [36] and the agent is controllable. The closed-loop dynamics for the i-th agent are now linear and given by(5)x˙it=vit+ρit,i∈1,…,N.The complete closed-loop dynamics for all the agents can be written as(6)x˙t=vt+ρtwhere v=[v1,…,vN]T∈RN.In a MAS, the consensus error of agenti with respect to its neighbors is defined as [2](7)eit=∑j∈Niaijxjt-xit,i∈1,…,N.The consensus error for the complete MAS can be expressed in a compact form as [2](8)et=e1t,…,eNtT=-Lxt.Problem statement: given a MAS with agents modeled as first-order nonlinear systems (3) and with an associated graph G, design a consensus protocol for each agent in the form ui=γ(ei,xi,t), such that the state of all the agents reaches a consensus value x∗ in apredefined time  tf from any initial state x(0), i.e., xi(t)→x∗,∀i={1,…,N}, as t→tf. Therefore, the consensus error (8) converges to zero at time tf.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].Atime base generator (TBG) is a continuous and differentiable time-dependent polynomial function h(t) described as [35](9)ht=τt·Cift∈0,tf0otherwisewhere τ(t)=[trtr-1…t1] is the time basis vector and C 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 tf, (10)ht=1,ift=00,ift≥tfh˙t=0,ift=0ort≥tf.In [35], a couple of procedures for calculating the coefficients C, 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., forρ=0N, 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 x∗ in a predefined time tf, from any initial state x(0). Consequently, the consensus error (8) converges to a neighborhood about the origin at tf. 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 x¯, provided that the MAS (4) is not perturbed and the initial state is known, (11)uit=1gixi-fixi+vi,i∈1,…,Nvi=h˙txi0-x¯where h(t) fulfills (10). In that work, it was demonstrated that consensus to x¯ is achieved in time tf and the resulting consensus error trajectory in the interval t∈[0,tf] is given by (12)e^t=hte0=-htLx0where e(0)=e0=[e1(0),…,eN(0)]T is the consensus error vector (8) at the initial condition x(0)=x0=x10,…,xN0T.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,(13)uit=1gixi-fixi+vi,i∈1,…,Nvi=h˙txi0-x¯+kfeit-htei0.This protocol guarantees stability during the transient behavior and also after the settling time, i.e., for anyt≥tf. 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 ei(0) to the origin at time tf, following a transient profile defined by the TBG as e^i(t)=h(t)ei(0); the second term is a feedback term that corrects any deviation of ei(t) from the transient profile e^i(t), 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 x¯, 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 asx^0=[x^1(0),…,x^N(0)]T, instead of the real initial state x0. In this way, (14)x^0=x0+dwhere d=[d1,…,dN]T is a vector of uncertainty in the initial state. From (14), the uncertain initial consensus error can be computed as (15)e^0=e^1t,…,e^NtT=-Lx^0.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 statex^0 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 kf∈R+ be a constant state-feedback gain. Then, the time-varying feedback control law (16)uit=1gixi-fixi+vi,i∈1,…,Nvi=-h˙te^i0+kfeit-hte^i0,with h(t) as in (10) and e^i(0) computed from the initial state estimate x^0, provides global asymptotic stability of the tracking error ξ(t)=e(t)-h(t)e^0. Therefore, assuming ρ(t)=0N, predefined-time convergence of the MAS (4) to a consensus value x∗ is achieved at tf from any initial state x0.Proof.
First, let us demonstrate the stability of the tracking error. The dynamics (3) with ρi(t)=0,∀i=1,…,N, under the control action (16), can be written in vector notation as (17)x˙t=-h˙te^0+kfet-hte^0.
Computinge˙(t) from (8) and using the linearized dynamics (17), we have(18)ξ˙t=e˙t-h˙te^0=-Lx˙t-h˙te^0=-Lx˙t+h˙tLx^0=-h˙tL2x^0-kfLξt+h˙tLx^0=-kfLξt+h˙tLx^0-Lx^0.
By usingh(0)=1, the initial tracking error is computed as (19)ξ0=e0-h0e^0=-Lx0+Lx0+d=Ld=SJS-1d,with L=SJS-1, where J∈CN×N is the Jordan form associated with L and S∈CN×N is a nonsingular matrix [39].
Letζ(t)=S-1ζ(t), then ζ(0)=JS-1d, and according to (2), it follows that the first entry of ζ(0) is null, i.e., ζ1(0)=0. Now, taking the time derivative of ζ(t) and using (18), we have (20)ζ˙t≜ζ˙1t,…,ζ˙NtT=S-1ξ˙t=S-1-kfLξt+h˙tLx^0-Lx^0=-kfS-1SJS-1Sζt+h˙tS-1SJS-1x^0-SJS-1x^0=-kfJζt+h˙tJS-1-JS-1x^0.
Letϕ(t)=ζ2t,…,ζNtT, and S-1=rY, where r∈R1×N and Y∈C(N-1)×N. Then, we can verify that (20) can be written as (21)ζ˙1t=0ϕ˙t=-kfJ^ϕt+Ftwhere F(t)=h˙(t)J^(Y-J^Y)x^(0) and J^∈C(N-1)×(N-1) is the Jordan matrix associated with L for λ2,…,λN, Re⁡(λi(L))>0,∀i={2,…,N}.
It can be shown that the solutions of the differential equations (21) are the following [44]: (22)ζ1t=ζ10=0ϕt=e-kfJ^tϕ0+∫0tekfJ^sFsds,t∈0,tfwhere e-kfJ^t∈C(N-1)×(N-1) is the exponential matrix. Then, it follows that (23)limt→+∞⁡ϕt=limt→+∞⁡ζ2t,…,ζNtT=0N-1,since the integral is a finite value due to h˙(t)=0,∀t≥tf, and -kfJ^ has eigenvalues with negative real parts [18, 45]. Notice that the convergence speed is determined by the dominant eigenvalue λ2 of L ([15]) and the control gain kf.
Sincelimt→+∞ζi(t)=0,∀i∈1,…,N, then (24)limt→+∞ξt=Slimt→+∞ζt=Slimt→+∞ζ1t,…,ζNtT=S0,…,0T=0N.
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 -kfJ^ is negative, ζ1(t)=0, and F(t) can be seen as a disturbance that becomes null for t≥tf. The stability of the tracking error implies that e(t) follows h(t)e^0; thus e(t) converges to the origin at the predefined-time tf. Any small final error in time t≥tf is corrected by the control input ui=(1/gi(xi))(-fi(xi)+kfei(t)), which is applied to each agent for t≥tf.
By using the propertyL1N=0N (in accordance to Lemma 5), the converge of the consensus error (8) to the origin implies (25)xt→xtf=x∗1Nast→tf,i.e.,(26)xit→x∗ast→tf,∀i=1,…,N.
That is, the system (4) achieves predefined-time convergence to a consensus value x∗ at tf using the consensus protocol (16).
Now, let us compute the final consensus value. First, definey(t)=γTx(t), where γ is defined in Lemma 5. For the two types of considered graphs, y(t) is an invariant quantity, since(27)y˙t=γTx˙t=γT-h˙te^0+kfet-hte^0=γTLh˙tx^0-kfxt-htx^0=0,∀xtwhere γTL=0NT by Lemma 5. Thus, (28)limt→tfyt=y0=γTx0
Moreover, from (25) it follows that (29)limt→tfyt=γTlimt→tfxt=γTx∗1N.
Then, for a connected undirected graph and by Lemma5 (γ=1N), it is obtained that (30)γTx∗1N=γTx0,x∗1NT1N=1NTx0,and solving for x∗, one has (31)x∗=∑i=1Nxi0N=Avex0.
Hence, the consensus valuex∗ is the average of the initial state x0 for connected undirected graphs. Now, for a directed graph with a directed spanning tree and by Lemma 5 (γT1N=1), it results in (32)x∗=γTx0γT1N=γTx0.
Therefore, the consensus value is given by (32) for a directed graph with a directed spanning tree.
Thus, system (4) with ρ(t)=0N achieves predefined-time convergence to a consensus value x∗ at tf in the presence of uncertainty in the initial state x0 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 vi in (16) is not sufficient to drive the consensus error trajectory to the TBG reference trajectory e^(t)=h(t)e^0 even for d=0N; consequently, system (4) under the first term of (16) does not reach predefined-time convergence to a consensus value x∗ at tf. In such a case, the final state is given by x(tf)=(IN-L)x0-Ld, where IN denotes the identity matrix of size N. 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 d in the initial state. In this sense, the compensation term for tracking must guarantee reaching the TBG trajectories before the desired convergence time tf. This means that the tracking control gain kf must be higher for a larger uncertainty in the initial state. Then, there is a limit on the values of d that can be managed by the consensus protocol.Remark 9.
Theorem7 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 vi(t) in (16) are smooth signals even at t=0, 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 Theorem7, 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.,ρ(t)=0N). 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 leaderxl(t)∈R in the network system, whose dynamics can be modeled as in [47] by (33)x˙lt=ultwhere ul(t)∈R is the leader’s control input. In this case, N agents are now the followers with dynamics (3). The consensus error for each follower is defined as [47](34)eift=∑j∈Niaijxjt-xit-bixit-xltwhere bi∈R represents the leader’s adjacency with bi>0 if agent i is a neighbor of the leader; otherwise bi=0,∀i={1,…,N}.The tracking error is given by(35)ξit=eift-hte^i0where h(t)e^i(0) is the TBG reference trajectory for the i-th and e^i(0) is computed from the initial state estimate x^0 as in (15). The tracking error will be used as the sliding surface; thus (36)sit=ξit=eift-hte^i0.Theorem 11.
Consider a MAS modeled as in (4) with a leader (33). For each agent i, define βi=(∑j∈Niaij+bi) and the sliding surface (36). Consider a function h(t) as in (10). There exist gains k1>0 and k2>ϱ(M), for some function ϱ(•) and a constant M>0, such that the following nonlinear controller defined for each agent (37)uit=βigi-1v¯i+vi,i∈1,…,Nv¯i=biul+∑j∈Niaijfj+gjuj-βifi-h˙te^i0vi=k1si1/2sign⁡si-wiw˙i=-k2sign⁡siachieves predefined-time convergence to the leader’s state xl at tf, allowing ρi(t)≠0, from any initial state xi(0).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 that(38)ξ˙it=e˙ift-h˙te^i0=∑j∈Niaijx˙jt-x˙it-bix˙it-x˙lt-h˙te^i0=-βifi+giui+∑j∈Niaijfj+gjuj+biul-βiρi+∑j∈Niaijρj-h˙te^i0.
Now, using (38) and plugging the control law (37) in the time derivative of the sliding surface (36), it results in (39)s˙i=ξ˙it=-k1si1/2sign⁡si+wi+ρdiw˙i=-k2sign⁡si,where ρdi=∑j∈Niaijρj-βiρi. Let zi=wi+ρdi; then the previous expression can be rewritten as (40)s˙i=-k1si1/2sign⁡si+ziz˙i=-k2sign⁡si+ρ˙di.
It has been proven in [46] that, for a bounded continuously differentiable disturbance, i.e., if ρdi<L and ρ˙di<M for some constants L>0, M>0, the second-order dynamics (40) converge globally to the origin (si=0,zi=0) in finite time in spite of the disturbance, if adequate positive control gains k1>0 and k2>ϱ(M) are used. Moreover, the remaining dynamics of the tracking error system are constrained to the sliding surface, such that si=s˙i=0, meaning that each agent eif(t) follows h(t)e^i(0). Consequently, the consensus error converges to zero in the predefined-time tf and the consensus value is given by the leader’s state xl [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 settf, considering that a small tf will result in large control inputs. Thus the maximum allowable input of each agent must be taken into consideration to set tf. 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.


## 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 x∗ in a predefined time tf, from any initial state x(0). Consequently, the consensus error (8) converges to a neighborhood about the origin at tf. 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 x¯, provided that the MAS (4) is not perturbed and the initial state is known, (11)uit=1gixi-fixi+vi,i∈1,…,Nvi=h˙txi0-x¯where h(t) fulfills (10). In that work, it was demonstrated that consensus to x¯ is achieved in time tf and the resulting consensus error trajectory in the interval t∈[0,tf] is given by (12)e^t=hte0=-htLx0where e(0)=e0=[e1(0),…,eN(0)]T is the consensus error vector (8) at the initial condition x(0)=x0=x10,…,xN0T.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,(13)uit=1gixi-fixi+vi,i∈1,…,Nvi=h˙txi0-x¯+kfeit-htei0.This protocol guarantees stability during the transient behavior and also after the settling time, i.e., for anyt≥tf. 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 ei(0) to the origin at time tf, following a transient profile defined by the TBG as e^i(t)=h(t)ei(0); the second term is a feedback term that corrects any deviation of ei(t) from the transient profile e^i(t), 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 x¯, 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 asx^0=[x^1(0),…,x^N(0)]T, instead of the real initial state x0. In this way, (14)x^0=x0+dwhere d=[d1,…,dN]T is a vector of uncertainty in the initial state. From (14), the uncertain initial consensus error can be computed as (15)e^0=e^1t,…,e^NtT=-Lx^0.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 statex^0 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 kf∈R+ be a constant state-feedback gain. Then, the time-varying feedback control law (16)uit=1gixi-fixi+vi,i∈1,…,Nvi=-h˙te^i0+kfeit-hte^i0,with h(t) as in (10) and e^i(0) computed from the initial state estimate x^0, provides global asymptotic stability of the tracking error ξ(t)=e(t)-h(t)e^0. Therefore, assuming ρ(t)=0N, predefined-time convergence of the MAS (4) to a consensus value x∗ is achieved at tf from any initial state x0.Proof.
First, let us demonstrate the stability of the tracking error. The dynamics (3) with ρi(t)=0,∀i=1,…,N, under the control action (16), can be written in vector notation as (17)x˙t=-h˙te^0+kfet-hte^0.
Computinge˙(t) from (8) and using the linearized dynamics (17), we have(18)ξ˙t=e˙t-h˙te^0=-Lx˙t-h˙te^0=-Lx˙t+h˙tLx^0=-h˙tL2x^0-kfLξt+h˙tLx^0=-kfLξt+h˙tLx^0-Lx^0.
By usingh(0)=1, the initial tracking error is computed as (19)ξ0=e0-h0e^0=-Lx0+Lx0+d=Ld=SJS-1d,with L=SJS-1, where J∈CN×N is the Jordan form associated with L and S∈CN×N is a nonsingular matrix [39].
Letζ(t)=S-1ζ(t), then ζ(0)=JS-1d, and according to (2), it follows that the first entry of ζ(0) is null, i.e., ζ1(0)=0. Now, taking the time derivative of ζ(t) and using (18), we have (20)ζ˙t≜ζ˙1t,…,ζ˙NtT=S-1ξ˙t=S-1-kfLξt+h˙tLx^0-Lx^0=-kfS-1SJS-1Sζt+h˙tS-1SJS-1x^0-SJS-1x^0=-kfJζt+h˙tJS-1-JS-1x^0.
Letϕ(t)=ζ2t,…,ζNtT, and S-1=rY, where r∈R1×N and Y∈C(N-1)×N. Then, we can verify that (20) can be written as (21)ζ˙1t=0ϕ˙t=-kfJ^ϕt+Ftwhere F(t)=h˙(t)J^(Y-J^Y)x^(0) and J^∈C(N-1)×(N-1) is the Jordan matrix associated with L for λ2,…,λN, Re⁡(λi(L))>0,∀i={2,…,N}.
It can be shown that the solutions of the differential equations (21) are the following [44]: (22)ζ1t=ζ10=0ϕt=e-kfJ^tϕ0+∫0tekfJ^sFsds,t∈0,tfwhere e-kfJ^t∈C(N-1)×(N-1) is the exponential matrix. Then, it follows that (23)limt→+∞⁡ϕt=limt→+∞⁡ζ2t,…,ζNtT=0N-1,since the integral is a finite value due to h˙(t)=0,∀t≥tf, and -kfJ^ has eigenvalues with negative real parts [18, 45]. Notice that the convergence speed is determined by the dominant eigenvalue λ2 of L ([15]) and the control gain kf.
Sincelimt→+∞ζi(t)=0,∀i∈1,…,N, then (24)limt→+∞ξt=Slimt→+∞ζt=Slimt→+∞ζ1t,…,ζNtT=S0,…,0T=0N.
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 -kfJ^ is negative, ζ1(t)=0, and F(t) can be seen as a disturbance that becomes null for t≥tf. The stability of the tracking error implies that e(t) follows h(t)e^0; thus e(t) converges to the origin at the predefined-time tf. Any small final error in time t≥tf is corrected by the control input ui=(1/gi(xi))(-fi(xi)+kfei(t)), which is applied to each agent for t≥tf.
By using the propertyL1N=0N (in accordance to Lemma 5), the converge of the consensus error (8) to the origin implies (25)xt→xtf=x∗1Nast→tf,i.e.,(26)xit→x∗ast→tf,∀i=1,…,N.
That is, the system (4) achieves predefined-time convergence to a consensus value x∗ at tf using the consensus protocol (16).
Now, let us compute the final consensus value. First, definey(t)=γTx(t), where γ is defined in Lemma 5. For the two types of considered graphs, y(t) is an invariant quantity, since(27)y˙t=γTx˙t=γT-h˙te^0+kfet-hte^0=γTLh˙tx^0-kfxt-htx^0=0,∀xtwhere γTL=0NT by Lemma 5. Thus, (28)limt→tfyt=y0=γTx0
Moreover, from (25) it follows that (29)limt→tfyt=γTlimt→tfxt=γTx∗1N.
Then, for a connected undirected graph and by Lemma5 (γ=1N), it is obtained that (30)γTx∗1N=γTx0,x∗1NT1N=1NTx0,and solving for x∗, one has (31)x∗=∑i=1Nxi0N=Avex0.
Hence, the consensus valuex∗ is the average of the initial state x0 for connected undirected graphs. Now, for a directed graph with a directed spanning tree and by Lemma 5 (γT1N=1), it results in (32)x∗=γTx0γT1N=γTx0.
Therefore, the consensus value is given by (32) for a directed graph with a directed spanning tree.
Thus, system (4) with ρ(t)=0N achieves predefined-time convergence to a consensus value x∗ at tf in the presence of uncertainty in the initial state x0 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 vi in (16) is not sufficient to drive the consensus error trajectory to the TBG reference trajectory e^(t)=h(t)e^0 even for d=0N; consequently, system (4) under the first term of (16) does not reach predefined-time convergence to a consensus value x∗ at tf. In such a case, the final state is given by x(tf)=(IN-L)x0-Ld, where IN denotes the identity matrix of size N. 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 d in the initial state. In this sense, the compensation term for tracking must guarantee reaching the TBG trajectories before the desired convergence time tf. This means that the tracking control gain kf must be higher for a larger uncertainty in the initial state. Then, there is a limit on the values of d that can be managed by the consensus protocol.Remark 9.
Theorem7 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 vi(t) in (16) are smooth signals even at t=0, 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 Theorem7, 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.,ρ(t)=0N). 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 leaderxl(t)∈R in the network system, whose dynamics can be modeled as in [47] by (33)x˙lt=ultwhere ul(t)∈R is the leader’s control input. In this case, N agents are now the followers with dynamics (3). The consensus error for each follower is defined as [47](34)eift=∑j∈Niaijxjt-xit-bixit-xltwhere bi∈R represents the leader’s adjacency with bi>0 if agent i is a neighbor of the leader; otherwise bi=0,∀i={1,…,N}.The tracking error is given by(35)ξit=eift-hte^i0where h(t)e^i(0) is the TBG reference trajectory for the i-th and e^i(0) is computed from the initial state estimate x^0 as in (15). The tracking error will be used as the sliding surface; thus (36)sit=ξit=eift-hte^i0.Theorem 11.
Consider a MAS modeled as in (4) with a leader (33). For each agent i, define βi=(∑j∈Niaij+bi) and the sliding surface (36). Consider a function h(t) as in (10). There exist gains k1>0 and k2>ϱ(M), for some function ϱ(•) and a constant M>0, such that the following nonlinear controller defined for each agent (37)uit=βigi-1v¯i+vi,i∈1,…,Nv¯i=biul+∑j∈Niaijfj+gjuj-βifi-h˙te^i0vi=k1si1/2sign⁡si-wiw˙i=-k2sign⁡siachieves predefined-time convergence to the leader’s state xl at tf, allowing ρi(t)≠0, from any initial state xi(0).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 that(38)ξ˙it=e˙ift-h˙te^i0=∑j∈Niaijx˙jt-x˙it-bix˙it-x˙lt-h˙te^i0=-βifi+giui+∑j∈Niaijfj+gjuj+biul-βiρi+∑j∈Niaijρj-h˙te^i0.
Now, using (38) and plugging the control law (37) in the time derivative of the sliding surface (36), it results in (39)s˙i=ξ˙it=-k1si1/2sign⁡si+wi+ρdiw˙i=-k2sign⁡si,where ρdi=∑j∈Niaijρj-βiρi. Let zi=wi+ρdi; then the previous expression can be rewritten as (40)s˙i=-k1si1/2sign⁡si+ziz˙i=-k2sign⁡si+ρ˙di.
It has been proven in [46] that, for a bounded continuously differentiable disturbance, i.e., if ρdi<L and ρ˙di<M for some constants L>0, M>0, the second-order dynamics (40) converge globally to the origin (si=0,zi=0) in finite time in spite of the disturbance, if adequate positive control gains k1>0 and k2>ϱ(M) are used. Moreover, the remaining dynamics of the tracking error system are constrained to the sliding surface, such that si=s˙i=0, meaning that each agent eif(t) follows h(t)e^i(0). Consequently, the consensus error converges to zero in the predefined-time tf and the consensus value is given by the leader’s state xl [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 settf, considering that a small tf will result in large control inputs. Thus the maximum allowable input of each agent must be taken into consideration to set tf. 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 fi(xi)=xi2 for agents 1 to 4, fi(xi)=sin⁡(xi) for agents 5 to 8, and gi(xi)=5 for all the 8 agents. The convergence time is preset to tf=5 seconds. For this, the TBG is described by the function h(t)=2t/tf3-3t/tf2+1, which fulfills (9), i.e., h(0)=1, h(5)=0, and h˙(0)=h˙(5)=0. 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 h(t) and h˙(t) 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 Figure1 (obtained from [30]), denoted as G1 and G2, where G1 is a connected undirected graph and G2 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.Figure 1
Communication graphs. Left: undirected graphG1. Right: directed graph G2.The initial states of the eight agents arex0=-0.61, 3.52, -1.07, 0.73, 1.48, 0.19, 1.97, 0.72T. The uncertainties in the initial states are set to be d=0.22, 0.15, -0.39, 0.27, 0.34, 0.21, -0.36, -0.09T.


### 4.1. Simulation of the Consensus Protocol Based on the Error Feedback

This subsection is devoted to showing the performance of the TBG protocol (16). Figures 2 and 3 show the transient behavior of the MAS under (16) with kf=100, for G1 and G2, respectively. For the case shown in Figure 2, perfect knowledge of the initial conditions is assumed, while in Figure 3, there exists uncertainty in the initial condition knowledge, given by d. In both cases the agent’s state (top graphs in the figures) reaches consensus at the predefined 5 seconds.Figure 2
Predefined-time TBG-tracking controller (16) for G1. The convergence time is preset to 5 sec., and the consensus value is ∑i=1Nxi(0)/N=0.8676.Figure 3
Predefined-time TBG-tracking controller (16) for G2 and uncertainty in the initial state. In the error trajectories, the continuous lines represent the evolution of the errors whereas the TBG reference is drawn with dash line. The convergence time is preset to 5 sec., and the consensus value is γTx0=0.0946.It can be seen in Figure2 (middle) that the auxiliary control inputs vi(t) generated by the control protocol (16) are smooth signals even at t=0. This is due to the properties of the TBG, which yields that each vi starts at zero and returns to zero at time tf. This is not the case for the control inputs ui, since they include the nonlinearities that must be cancelled. In the case where the uncertainty d in the initial state is considered (Figure 3), the second term of vi in (16) is not initially null to compensate the deviation from the reference given by the TBG. It can be seen in Figure 3 (middle and bottom) that the consensus errors do not initiate over the references and consequently the auxiliary controls are not null at t=0.


### 4.2. Comparison with Others Approaches

In this subsection, first, we compare the proposed control law (16) with other predefined-time control protocols [30–32]. As described in Section 1, to the authors’ knowledge, these are the existing approaches that solve the problem of predefined-time convergence. Each one of these works refers to their proposal with a different adjective: preset-time consensus [30], prescribed-time consensus [31], and specified-time consensus [32]. We can see in Figures 4–6 that consensus is achieved by using any of these approaches. We have to say that the control protocols of [30, 31] are very similar; both of them are based on the use of a time-varying control gain. Between their similarities, we can see in Figures 4 and 5 the large initial value of the auxiliary control inputs. This is also the case for the specified-time consensus [32] of Figure 6. Notice that the auxiliary control inputs of the TBG-based controller, shown in Figure 2, started in zero and were smoother and of lower magnitude compared to the large initial auxiliary control effort of the other predefined-time consensus approaches.Figure 4
Preset-time controller [30] for G1. The convergence time is preset to 5 sec., and the consensus value is ∑i=1Nxi(0)/N=0.8676.Figure 5
Prescribed-time controller [31] for G2. The convergence time is present to 5 sec., and the consensus value is 0.0946.Figure 6
Specified-time controller [32] for G1. The convergence time is present to 5 sec., and the consensus value is 0.5196.Additionally, we can see that the specified-time consensus protocol in Figure6 generates discontinuities in the auxiliary inputs along the time, according to its planned switching strategy. Moreover, although this consensus approach was tested on the undirected graph G1, the consensus value is not the average of the initial state as one could expect.Following the comparison, we evaluate the TBG controller (16) and different finite-time, fixed-time, and predefined-time consensus protocols reported in the literature, by applying the protocols to the MAS with the communication topologies G1 and G2. In particular, the protocols in [20, 48] ensure finite-time convergence, and the one proposed in [27] guarantees fixed-time convergence. The predefined-time protocols [30–32] and the proposed TBG-tracking controller (16) achieve predefined-time convergence. The finite-time and fixed-time controllers were tuned to achieve a similar convergence time around the 5 seconds of predefined time, from the initial condition x0. Then, several simulations were performed for different initial states of the form αx0, where α was varying in such a way that the mean of αx0 ranges from -50 to 50. The same control gains were used in all the experiments. Finally, for every experiment, the convergence time of the system, defined as the time at which e<1×10-4, was measured; similarly, the maximum absolute value of the auxiliary control input v was recorded.The results are shown in Figures7 and 8, where it can be observed that the proposed TBG-based control (16) and the other predefined-time controllers [30–32] are able to maintain the same convergence time for all the initial conditions. Furthermore, note that the maximum auxiliary control efforts were lower for both the TBG-based control and the finite-time control [20]. The last control law is of the form ksign⁡(•). In particular, the consensus protocols [30, 31] explicitly include a parameter that must be set accordingly to an eigenvalue of the Laplacian matrix, which depends on the connectivity of the graph considered. This is not the case for the proposed control protocols. Therefore, the simulations show that the proposed TBG protocol (16) represents an advantage not only in the possibility to define a settling time but also in providing smooth and efficient control actions.Figure 7
Comparison of the TBG-tracking controller (16) with other approaches for G1. Top: settling time as a function of the initial condition x0. Bottom: maximum value of the auxiliary control input as a function of the initial condition x0.Figure 8
Comparison of the TBG-tracking controller (16) with other approaches for G2. Top: settling time as a function of the initial condition x0. Bottom: maximum value of the auxiliary control input as a function of the initial condition x0.


### 4.3. Robust Predefined-Time Consensus

All the previous simulations considered ideal dynamics in (3), i.e., ρi(t)=0 for all the agents. In fact, the TBG-tracking controller (16) is not able to deal with matched bounded disturbances. For instance, Figure 9 shows the MAS with the topology G2 under this protocol and matching perturbations ρi(t)=αi(1+1sin⁡(5t)), with αi randomly selected in (0,0.5). It can be observed that the state of the MAS does not establish in a consensus value due to the perturbation. A similar behavior can be obtained with the other existing predefined-time consensus protocols [30–32].Figure 9
Predefined-time TBG-tracking controller (16), which is not robust for perturbed dynamics and G2.In order to deal with the described matching perturbationsρi(t)=αi(1+1sin⁡(5t)), the proposed robust predefined-time TBG controller (37) has to be used. For this, a leader is defined with dynamics (33), having communication only with the first follower agent, i.e., b1=1 and bi=0,∀i=2,…,8. The gains of the robust controller (37) were set as k1=5 and k2=5; and the leader’s state was maintained constant and equal to xl(t)=-5 by setting its control input as ul=0. Figures 10 and 11 show the results obtained when the communication between the followers is described by G1 and G2, respectively. Notice that the error trajectories are driven to zero in the predefined time tf=5, while the control input keeps oscillating after tf in order to reject the disturbance ρ(t).Figure 10
Robust predefined-time TBG controller (37) for perturbed systems and G1. In the error trajectories, the continuous lines represent the evolution of the errors whereas the TBG reference is drawn with dash line. The convergence time is preset to 5 sec.Figure 11
Robust predefined-time TBG controller (37) for perturbed systems and G2. In the error trajectories, the continuous lines represent the evolution of the errors whereas the TBG reference is drawn with dash line. The convergence time is preset to 5 sec.


## 4.1. Simulation of the Consensus Protocol Based on the Error Feedback

This subsection is devoted to showing the performance of the TBG protocol (16). Figures 2 and 3 show the transient behavior of the MAS under (16) with kf=100, for G1 and G2, respectively. For the case shown in Figure 2, perfect knowledge of the initial conditions is assumed, while in Figure 3, there exists uncertainty in the initial condition knowledge, given by d. In both cases the agent’s state (top graphs in the figures) reaches consensus at the predefined 5 seconds.Figure 2
Predefined-time TBG-tracking controller (16) for G1. The convergence time is preset to 5 sec., and the consensus value is ∑i=1Nxi(0)/N=0.8676.Figure 3
Predefined-time TBG-tracking controller (16) for G2 and uncertainty in the initial state. In the error trajectories, the continuous lines represent the evolution of the errors whereas the TBG reference is drawn with dash line. The convergence time is preset to 5 sec., and the consensus value is γTx0=0.0946.It can be seen in Figure2 (middle) that the auxiliary control inputs vi(t) generated by the control protocol (16) are smooth signals even at t=0. This is due to the properties of the TBG, which yields that each vi starts at zero and returns to zero at time tf. This is not the case for the control inputs ui, since they include the nonlinearities that must be cancelled. In the case where the uncertainty d in the initial state is considered (Figure 3), the second term of vi in (16) is not initially null to compensate the deviation from the reference given by the TBG. It can be seen in Figure 3 (middle and bottom) that the consensus errors do not initiate over the references and consequently the auxiliary controls are not null at t=0.


## 4.2. Comparison with Others Approaches

In this subsection, first, we compare the proposed control law (16) with other predefined-time control protocols [30–32]. As described in Section 1, to the authors’ knowledge, these are the existing approaches that solve the problem of predefined-time convergence. Each one of these works refers to their proposal with a different adjective: preset-time consensus [30], prescribed-time consensus [31], and specified-time consensus [32]. We can see in Figures 4–6 that consensus is achieved by using any of these approaches. We have to say that the control protocols of [30, 31] are very similar; both of them are based on the use of a time-varying control gain. Between their similarities, we can see in Figures 4 and 5 the large initial value of the auxiliary control inputs. This is also the case for the specified-time consensus [32] of Figure 6. Notice that the auxiliary control inputs of the TBG-based controller, shown in Figure 2, started in zero and were smoother and of lower magnitude compared to the large initial auxiliary control effort of the other predefined-time consensus approaches.Figure 4
Preset-time controller [30] for G1. The convergence time is preset to 5 sec., and the consensus value is ∑i=1Nxi(0)/N=0.8676.Figure 5
Prescribed-time controller [31] for G2. The convergence time is present to 5 sec., and the consensus value is 0.0946.Figure 6
Specified-time controller [32] for G1. The convergence time is present to 5 sec., and the consensus value is 0.5196.Additionally, we can see that the specified-time consensus protocol in Figure6 generates discontinuities in the auxiliary inputs along the time, according to its planned switching strategy. Moreover, although this consensus approach was tested on the undirected graph G1, the consensus value is not the average of the initial state as one could expect.Following the comparison, we evaluate the TBG controller (16) and different finite-time, fixed-time, and predefined-time consensus protocols reported in the literature, by applying the protocols to the MAS with the communication topologies G1 and G2. In particular, the protocols in [20, 48] ensure finite-time convergence, and the one proposed in [27] guarantees fixed-time convergence. The predefined-time protocols [30–32] and the proposed TBG-tracking controller (16) achieve predefined-time convergence. The finite-time and fixed-time controllers were tuned to achieve a similar convergence time around the 5 seconds of predefined time, from the initial condition x0. Then, several simulations were performed for different initial states of the form αx0, where α was varying in such a way that the mean of αx0 ranges from -50 to 50. The same control gains were used in all the experiments. Finally, for every experiment, the convergence time of the system, defined as the time at which e<1×10-4, was measured; similarly, the maximum absolute value of the auxiliary control input v was recorded.The results are shown in Figures7 and 8, where it can be observed that the proposed TBG-based control (16) and the other predefined-time controllers [30–32] are able to maintain the same convergence time for all the initial conditions. Furthermore, note that the maximum auxiliary control efforts were lower for both the TBG-based control and the finite-time control [20]. The last control law is of the form ksign⁡(•). In particular, the consensus protocols [30, 31] explicitly include a parameter that must be set accordingly to an eigenvalue of the Laplacian matrix, which depends on the connectivity of the graph considered. This is not the case for the proposed control protocols. Therefore, the simulations show that the proposed TBG protocol (16) represents an advantage not only in the possibility to define a settling time but also in providing smooth and efficient control actions.Figure 7
Comparison of the TBG-tracking controller (16) with other approaches for G1. Top: settling time as a function of the initial condition x0. Bottom: maximum value of the auxiliary control input as a function of the initial condition x0.Figure 8
Comparison of the TBG-tracking controller (16) with other approaches for G2. Top: settling time as a function of the initial condition x0. Bottom: maximum value of the auxiliary control input as a function of the initial condition x0.


## 4.3. Robust Predefined-Time Consensus

All the previous simulations considered ideal dynamics in (3), i.e., ρi(t)=0 for all the agents. In fact, the TBG-tracking controller (16) is not able to deal with matched bounded disturbances. For instance, Figure 9 shows the MAS with the topology G2 under this protocol and matching perturbations ρi(t)=αi(1+1sin⁡(5t)), with αi randomly selected in (0,0.5). It can be observed that the state of the MAS does not establish in a consensus value due to the perturbation. A similar behavior can be obtained with the other existing predefined-time consensus protocols [30–32].Figure 9
Predefined-time TBG-tracking controller (16), which is not robust for perturbed dynamics and G2.In order to deal with the described matching perturbationsρi(t)=αi(1+1sin⁡(5t)), the proposed robust predefined-time TBG controller (37) has to be used. For this, a leader is defined with dynamics (33), having communication only with the first follower agent, i.e., b1=1 and bi=0,∀i=2,…,8. The gains of the robust controller (37) were set as k1=5 and k2=5; and the leader’s state was maintained constant and equal to xl(t)=-5 by setting its control input as ul=0. Figures 10 and 11 show the results obtained when the communication between the followers is described by G1 and G2, respectively. Notice that the error trajectories are driven to zero in the predefined time tf=5, while the control input keeps oscillating after tf in order to reject the disturbance ρ(t).Figure 10
Robust predefined-time TBG controller (37) for perturbed systems and G1. In the error trajectories, the continuous lines represent the evolution of the errors whereas the TBG reference is drawn with dash line. The convergence time is preset to 5 sec.Figure 11
Robust predefined-time TBG controller (37) for perturbed systems and G2. In the error trajectories, the continuous lines represent the evolution of the errors whereas the TBG reference is drawn with dash line. The convergence time is preset to 5 sec.


## 5. Conclusions

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 [4, 5].



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*Source: 1957070-2018-10-09.xml*