# Delayed Consensus Problem for Single and Double Integrator Systems

Authors: Martín Velasco-Villa; Josué Heras-Godínez; José Alejandro Vázquez-Santacruz; Varinia Fragoso-Rubio
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This work deals with the analysis of the consensus problem for networks of agents constituted by single and double integrator systems. It is assumed that the communication among agents is affected by a constant time-delay. Previous and numerous analysis of the problem shows that the maximum communication time-delay that can be introduced to the network without affecting the consensus of the group of the agents depends on the considered topology. In this work, a control scheme that is based on the estimation of future states of the agents and that allows increasing the magnitude of a possible time-delay affecting the communication channels is proposed. How the proposed delay compensation strategy is independent of the network topology in the sense that the maximum allowable time-delay that could be supported by the network depends on a design parameter and not on the maximum eigenvalue of the corresponding Laplacian matrix is shown. It is formally proven that, under the proposed prediction scheme, the consensus of the group can be achieved by improving the maximum time-delay bounds previously reported in the literature. Numerical simulations show the effectiveness of the proposed solution.


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## Body

## 1. Introduction

In the last years, the study of multiagent systems has deserved an increasing attention of the control community. These systems have been involved in many practical applications in everyday life and industry, as the case of the formation problem of unnamed aerial vehicles (UAV) [1], the control of a group of satellites [2], or formation control vehicles [3–5]. Some other applications are related to distributed sensors; automatic vehicular control; or analysis of communication networks [6, 7].A common problem in the synchronization of multiagent systems is the consensus problem [8], which is referred to the control and coordination of a set of agents within a communication network. In [3], the importance of the Laplacian of the network in the solution of the consensus problem for a set of agents with linear dynamics is shown. The problem has also been addressed in [8–11] getting the asymptotic convergence of the agent into the set.In an attempt to achieve consensus, an additional challenge arises when the agents coordination involves time-delays in the communication network which governs the agents. In control systems, time-delays frequently appear due to phenomena as material transportation, recycle loops, or even the approximation of high order systems by means of reduced order dynamics [12]. Several solutions have been proposed when dealing with stable processes with time-delay; in the linear case, when the time-delay is located at the input or output of the system, a common practice consists on eliminating the effect of the delayed signal by means of an adequate feedback [13–15].In general, the consensus problem has been commonly treated by considering that the agents are free of delays. The problem has been properly described in [7, 16, 17] and a solution has been clearly stated. The consensus problem has been also extended to the case of bounded inputs in [18]. On the other hand, the consideration of time-delay has been studied under several communication schemes, for example, by assuming that the agents are single integrators and that the time-delay is constant. Under these circumstances, an initial analysis that states the maximum time-delay that can be supported by the network is given in [9]. Another scheme, proposed in [19], assumes that the delays are induced only by the neighbor agents and that the local variables are free of delay. In [20], the problem is analyzed by considering that the delay exists on the position variables but not in the velocity of the neighbor agents; the same conditions are imposed in [21] where the velocity error is not used on the control law. By considering the local variables of the agents free of delays, in [22], the problem is studied in a discrete-time context, as well as [23–25]. A delayed position and velocity feedback is considered in [16], while the consensus problem is analyzed in [26] under the assumption of fixed topologies and in [27] with both fixed and switched topologies.Specific techniques are used to prove stability of a set of agents, for example, a frequency approach has been addressed in [15, 28, 29] by means of the Nyquist criterion, even for the presence of disturbance and noise as in [23, 30]. Also, the problem is not only addressed by considering a constant delay, but the time varying delay is also considered in works as [27, 31]. In addition, some strategies based on artificial neural networks are addressed in [32].In particular, the protocol adopted in [9], for the case of single integrator agents, guarantees the consensus, with their states asymptotically converging to the average of the initial conditions, instead of an arbitrary constant. The authors explicitly provide the largest time-delay that can be applied to the network without breaking the consensus. It is also shown that this largest delay is inversely proportional to the maximum eigenvalue of the Laplacian matrix induced by the network. In the case of double integrator agents with time-delays, in [21, 22, 26], it is also shown that the solution of the problem is related with the largest eigenvalue of the Laplacian matrix of the network.By considering the induced restriction by the Laplacian matrix eigenvalues to the solution of the delayed consensus problem, the objective of this work is to increase the largest time-delay that allows the solution of the problem. With this aim, a prediction-observer scheme that provides estimated future states of the agents that are used in the design of an adequate feedback is proposed, inspired by the solution of the free delay case originally given in [16].The rest of the work is organized as follows. In Section2, a brief recall of the graph theory and the Laplacian matrix is presented. In Section 3, the consensus problem is stated referring to the previous works reported in the literature. The single integrator case is addressed in Section 4 and the obtained result is generalized to the double integrator case in Section 5. Numerical simulations are presented in Section 6, showing the effectiveness of the proposed control scheme. Finally, some conclusions are given in Section 7.


## 2. Graph Theory and Laplacian Matrix

Graph theory studies structures of vertex, nodes, or point sets and edge or line sets [8, 33, 34]. Being of interest for this work, in what follows a summary of this topic is presented.


### 2.1. Graph Classes and Definitions

LetG={N,E} be an ordered graph where(i)
N is a nonempty node set defined as N={1,…,n};(ii)
E is an edge set of ordered node pairs, E⊆N×N;(iii)
an edge fromG is denoted as eij=(i,j);(iv)
the gradeN of a node equals the number of edges eij that belong to the node.Adirected graph is a pair (N,E) where the edge (i,j)∈E denotes that a j-agent can obtain information from an i-agent but not necessarily in the opposite way.For the edge(i,j), i is the parent node and j is the child one. In contrast to a directed graph, a pair of nodes in anundirected graph is unordered; that is, the edge (i,j) denotes that agents i and j can obtain information from each other. Adirected path is a sequence of edges in a directed graph of the form (i1,i2),(i2,i3),…. Anundirected path in an undirected graph is defined analogously. A cycle is a directed path that starts and ends at the same node. A directed graph isstrongly connected if there is a directed path from a node to any other node. An undirected graph is connected if there is an undirected path between every pair of distinct nodes. Theadjacency matrixA=[aij]∈Rn×n of a directed graph with node set N={1,…,n} is defined such that the weight aij is positive if (i,j)∈E and aij=0 if (i,j)∉E. If the weights are not relevant, then aij is set equal to 1 for all (i,j)∈E. A graph isbalanced if ∑j=1n‍aij=∑j=1n‍aji for all i. For an undirected graph, A is symmetric, and thus every undirected graph is balanced.


### 2.2. Laplacian Matrix

The Laplacian matrixL=[lij]∈Rn×n of a directed graph is given by lii=∑j≠i‍aij and lij=-aij for all i≠j.


#### 2.2.1. Laplacian Matrix Properties

(i)
l
i
j
≤
0
,
i
≠
j.
(ii)
The sum of the elements of each row is zero; Consider∑j=1n‍lij=0, i=1,…,n.
(iii)
For an undirected graph,L is symmetric.
(iv)
For directed and undirected graphs,0 is an eigenvalue of L, the sum of the elements of each row is zero, and it has an eigenvector 1n=[1,…,1]T, of dimension n×1, corresponding to the zero eigenvalue.
(v)
L is diagonally dominant and has no negative entries on its diagonal.
(vi)
All nonzero eigenvalues ofL are positive, while, in a directed graph, all nonzero eigenvalues of L have positive real parts. Therefore, all nonzero eigenvalues of -L have negative real parts.
(vii)
Zero is always an eigenvalue ofL with at least multiplicity one.
(viii)
For an undirected graph, zero is a simple eigenvalue ofL if and only if the undirected path is connected.
(ix)
For a directed graph, zero is a simple eigenvalue ofL if and only if the graph has a spanning tree.
(x)
If theith node of G is disconnected, then lij=lji=0 for all j.


## 2.1. Graph Classes and Definitions

LetG={N,E} be an ordered graph where(i)
N is a nonempty node set defined as N={1,…,n};(ii)
E is an edge set of ordered node pairs, E⊆N×N;(iii)
an edge fromG is denoted as eij=(i,j);(iv)
the gradeN of a node equals the number of edges eij that belong to the node.Adirected graph is a pair (N,E) where the edge (i,j)∈E denotes that a j-agent can obtain information from an i-agent but not necessarily in the opposite way.For the edge(i,j), i is the parent node and j is the child one. In contrast to a directed graph, a pair of nodes in anundirected graph is unordered; that is, the edge (i,j) denotes that agents i and j can obtain information from each other. Adirected path is a sequence of edges in a directed graph of the form (i1,i2),(i2,i3),…. Anundirected path in an undirected graph is defined analogously. A cycle is a directed path that starts and ends at the same node. A directed graph isstrongly connected if there is a directed path from a node to any other node. An undirected graph is connected if there is an undirected path between every pair of distinct nodes. Theadjacency matrixA=[aij]∈Rn×n of a directed graph with node set N={1,…,n} is defined such that the weight aij is positive if (i,j)∈E and aij=0 if (i,j)∉E. If the weights are not relevant, then aij is set equal to 1 for all (i,j)∈E. A graph isbalanced if ∑j=1n‍aij=∑j=1n‍aji for all i. For an undirected graph, A is symmetric, and thus every undirected graph is balanced.


## 2.2. Laplacian Matrix

The Laplacian matrixL=[lij]∈Rn×n of a directed graph is given by lii=∑j≠i‍aij and lij=-aij for all i≠j.


### 2.2.1. Laplacian Matrix Properties

(i)
l
i
j
≤
0
,
i
≠
j.
(ii)
The sum of the elements of each row is zero; Consider∑j=1n‍lij=0, i=1,…,n.
(iii)
For an undirected graph,L is symmetric.
(iv)
For directed and undirected graphs,0 is an eigenvalue of L, the sum of the elements of each row is zero, and it has an eigenvector 1n=[1,…,1]T, of dimension n×1, corresponding to the zero eigenvalue.
(v)
L is diagonally dominant and has no negative entries on its diagonal.
(vi)
All nonzero eigenvalues ofL are positive, while, in a directed graph, all nonzero eigenvalues of L have positive real parts. Therefore, all nonzero eigenvalues of -L have negative real parts.
(vii)
Zero is always an eigenvalue ofL with at least multiplicity one.
(viii)
For an undirected graph, zero is a simple eigenvalue ofL if and only if the undirected path is connected.
(ix)
For a directed graph, zero is a simple eigenvalue ofL if and only if the graph has a spanning tree.
(x)
If theith node of G is disconnected, then lij=lji=0 for all j.


## 2.2.1. Laplacian Matrix Properties

(i)
l
i
j
≤
0
,
i
≠
j.
(ii)
The sum of the elements of each row is zero; Consider∑j=1n‍lij=0, i=1,…,n.
(iii)
For an undirected graph,L is symmetric.
(iv)
For directed and undirected graphs,0 is an eigenvalue of L, the sum of the elements of each row is zero, and it has an eigenvector 1n=[1,…,1]T, of dimension n×1, corresponding to the zero eigenvalue.
(v)
L is diagonally dominant and has no negative entries on its diagonal.
(vi)
All nonzero eigenvalues ofL are positive, while, in a directed graph, all nonzero eigenvalues of L have positive real parts. Therefore, all nonzero eigenvalues of -L have negative real parts.
(vii)
Zero is always an eigenvalue ofL with at least multiplicity one.
(viii)
For an undirected graph, zero is a simple eigenvalue ofL if and only if the undirected path is connected.
(ix)
For a directed graph, zero is a simple eigenvalue ofL if and only if the graph has a spanning tree.
(x)
If theith node of G is disconnected, then lij=lji=0 for all j.


## 3. Statement of the Problem

The main idea of the consensus protocol is to impose a similar dynamic to the state of each agent. If the communication channels in the network allow continuous flow information, then the evolution of each agent can be modeled, for instance, by means of a differential equation. In this work it is considered fixed information exchange topologies with Laplacian matrices with a single zero eigenvalue; that is, directed graphs with an spanning tree or undirected connected graphs [34] are considered.Definition 1.
The consensus problem is solved for a group of agents if, for all initial conditionxi(0), it is satisfied as follows: (1)xi(t)-xj(t)⟶0as t→∞, for all i,j=1,…,n.


### 3.1. Consensus for Single Integrator Agents

Consider the class of single-input single-output agents of the form(2)x˙i(t)=ui(t),for i=1,2,…,n, with xi∈R being the state associated with the ith agent while ui∈R is the corresponding input signal. Define, for a group of n agents, x=[x1,…,xn]T∈Rn and u=[u1,…,un]T∈Rn.The consensus protocol for the case of a fixed [4, 8, 9] or variant [9] topology in the case of agents (2) is given by(3)ui(t)=-∑j=1n‍aijxit-xjt.The closed-loop (2)-(3) produces(4)x˙t=-Lxt,where L∈Rn×n is the Laplacian matrix induced by the considered topology.In this work, it will be assumed that, for a fixed topology of a directed graph with a spanning treeG={N,E}, there exists a constant time-delay τij>0 affecting the edge eij∈E; in particular, it will be considered that the time-delay is equal for all the communication channels: τij=τ. Under the effects of communication time-delays, the consensus protocol (3) is modified to the delayed state causal feedback [9]: (5)ui(t)=-∑j=1n‍aijxit-τ-xjt-τthat produces the closed-loop system:(6)x˙(t)=-Lx(t-τ),where x∈Rn and xi(t-τ) is the state of the ith agent delayed τ units of time.Remark 2.
It should be noticed that, under the assumption thatτij=τ1>0, the delayed consensus problem can be stated by considering a set of delayed agents:(7)x˙i(t)=uit-τ1with a consensus protocol given in (3). In this form, agents (7), explicitly describe the effects of the communication channels. Notice also that, in the case that protocol (5) is applied to agents (7), a closed-loop system is obtained in the form(8)x˙(t)=-Lx(t-τ2),for τ2=τ+τ1. Therefore, the solution for the delayed consensus problem is equivalent for system (6) or (8).Based on the previous remark, the consensus problem will be stated in this work for a set of delayed agents(9)x˙i(t)=uit-τ.By considering a fixed topology, undirected and connected graphs, the consensus problem for agents (2) and protocol (5) (equivalently, agents (9) and protocol (3)) is solved, if and only if [9] (10)0≤τ<π2σmax⁡(L),where L is the Laplacian of the communication graph and σmax⁡(L) is the largest eigenvalue of L.


### 3.2. Consensus for Double Integrator Agents

As in the case of integrator agents, it will be assumed that there exist constant time-delaysτij=τ (corresponding to the edge eij∈E) affecting the communications channels. According to Remark 2, the effects of communication time-delays can be included on the agent dynamics in the form (11)x˙i1t=xi2t,x˙i2t=uit-τ,for i=1,2,…,n with xi1, xi2∈R being the state of the ith agent and ui∈R the corresponding input signal. τ>0 is the time-delay affecting the ith input signal.The consensus for system (11) is achieved according to Definition 1; that is, xi1(t)→xj1(t) and xi2(t)→xj2(t) when t→∞, for all xi1(0) and xi2(0).System (11) can be rewritten as(12)x˙1t=x2t,x˙2(t)=ut-τ,where x1=x11⋯xn1T, x2=x12⋯xn2T, and u=u1⋯unT.For the free delay case,τ=0 in (11), the solution of the consensus problem in the case of a directed graph containing a spanning tree is obtained by means of the state feedback [7, 18], (13)ui=-∑j=1n‍aijxi1-xj1+γxi2-xj2,for sufficiently large positive γ∈R, producing a closed-loop system with exactly two zero poles and all the remaining poles with negative real part.For a fixed undirected topology having a spanning tree, in [26], a solution to the delayed consensus problem (in the case that τij=τ) for double integrator agents is presented by considering a modification of the causal feedback (13) in the form(14)uit=-∑j=1n‍aijk1xi1t-xj1thhhhhhhhhxi1t-xj1t+k2xi2(t)-xj2(t),with k1 and k2 as positive constants. This solution is stated for a maximum time-delay τ∗ such that τ<τ∗, with(15)τ∗≤k2arctanζ1-ζk1ζ,where ζ is the positive root of(16)ζ3+1+4k1k22σmax⁡(L)ζ2+4k1k22σmax⁡(L)-1ζ-1=0.It is important to point out that the existing solutions for the delayed consensus problem for single (9) and double integrators (11) are based on the use of causal feedbacks of the forms (3) and (13) synthesized at actual time t producing time-delayed closed-loop dynamics.Suppose now that protocols (3) and (13) can be designed in terms of the future value, τ units of time, of the agents states, in the form(17)ui(t)=-∑j=1n‍aijxit+τ-xjt+τfor the single integrator case and (18)uit=-∑j=0n‍aijxi1t+τ-xj1t+τhhhhhhhhhh+γxi2t+τ-xj2t+τfor the double integrator agents.It is easy to see that protocols (17) and (18) would solve the delayed consensus problem producing closed-loop systems free of time-delay for the single and double integrator cases. Notice that the convergence properties in the case of system (9)–(17) or (11)–(18) would be equivalent to the ones of the free-delay cases [4, 7–9]. It should be highlighted that protocols (17) and (18) require future values of the agents states that prevent their practical implementation due to their noncausal nature.The main objective of this work is to propose estimated state protocols of the forms (17) and (18) where instead of the actual future states xi(t+τ) and xijt+τ, the use of future estimated states x^i(t+τ) and x^ijt+τ obtained by an adequate prediction observer is considered.


## 3.1. Consensus for Single Integrator Agents

Consider the class of single-input single-output agents of the form(2)x˙i(t)=ui(t),for i=1,2,…,n, with xi∈R being the state associated with the ith agent while ui∈R is the corresponding input signal. Define, for a group of n agents, x=[x1,…,xn]T∈Rn and u=[u1,…,un]T∈Rn.The consensus protocol for the case of a fixed [4, 8, 9] or variant [9] topology in the case of agents (2) is given by(3)ui(t)=-∑j=1n‍aijxit-xjt.The closed-loop (2)-(3) produces(4)x˙t=-Lxt,where L∈Rn×n is the Laplacian matrix induced by the considered topology.In this work, it will be assumed that, for a fixed topology of a directed graph with a spanning treeG={N,E}, there exists a constant time-delay τij>0 affecting the edge eij∈E; in particular, it will be considered that the time-delay is equal for all the communication channels: τij=τ. Under the effects of communication time-delays, the consensus protocol (3) is modified to the delayed state causal feedback [9]: (5)ui(t)=-∑j=1n‍aijxit-τ-xjt-τthat produces the closed-loop system:(6)x˙(t)=-Lx(t-τ),where x∈Rn and xi(t-τ) is the state of the ith agent delayed τ units of time.Remark 2.
It should be noticed that, under the assumption thatτij=τ1>0, the delayed consensus problem can be stated by considering a set of delayed agents:(7)x˙i(t)=uit-τ1with a consensus protocol given in (3). In this form, agents (7), explicitly describe the effects of the communication channels. Notice also that, in the case that protocol (5) is applied to agents (7), a closed-loop system is obtained in the form(8)x˙(t)=-Lx(t-τ2),for τ2=τ+τ1. Therefore, the solution for the delayed consensus problem is equivalent for system (6) or (8).Based on the previous remark, the consensus problem will be stated in this work for a set of delayed agents(9)x˙i(t)=uit-τ.By considering a fixed topology, undirected and connected graphs, the consensus problem for agents (2) and protocol (5) (equivalently, agents (9) and protocol (3)) is solved, if and only if [9] (10)0≤τ<π2σmax⁡(L),where L is the Laplacian of the communication graph and σmax⁡(L) is the largest eigenvalue of L.


## 3.2. Consensus for Double Integrator Agents

As in the case of integrator agents, it will be assumed that there exist constant time-delaysτij=τ (corresponding to the edge eij∈E) affecting the communications channels. According to Remark 2, the effects of communication time-delays can be included on the agent dynamics in the form (11)x˙i1t=xi2t,x˙i2t=uit-τ,for i=1,2,…,n with xi1, xi2∈R being the state of the ith agent and ui∈R the corresponding input signal. τ>0 is the time-delay affecting the ith input signal.The consensus for system (11) is achieved according to Definition 1; that is, xi1(t)→xj1(t) and xi2(t)→xj2(t) when t→∞, for all xi1(0) and xi2(0).System (11) can be rewritten as(12)x˙1t=x2t,x˙2(t)=ut-τ,where x1=x11⋯xn1T, x2=x12⋯xn2T, and u=u1⋯unT.For the free delay case,τ=0 in (11), the solution of the consensus problem in the case of a directed graph containing a spanning tree is obtained by means of the state feedback [7, 18], (13)ui=-∑j=1n‍aijxi1-xj1+γxi2-xj2,for sufficiently large positive γ∈R, producing a closed-loop system with exactly two zero poles and all the remaining poles with negative real part.For a fixed undirected topology having a spanning tree, in [26], a solution to the delayed consensus problem (in the case that τij=τ) for double integrator agents is presented by considering a modification of the causal feedback (13) in the form(14)uit=-∑j=1n‍aijk1xi1t-xj1thhhhhhhhhxi1t-xj1t+k2xi2(t)-xj2(t),with k1 and k2 as positive constants. This solution is stated for a maximum time-delay τ∗ such that τ<τ∗, with(15)τ∗≤k2arctanζ1-ζk1ζ,where ζ is the positive root of(16)ζ3+1+4k1k22σmax⁡(L)ζ2+4k1k22σmax⁡(L)-1ζ-1=0.It is important to point out that the existing solutions for the delayed consensus problem for single (9) and double integrators (11) are based on the use of causal feedbacks of the forms (3) and (13) synthesized at actual time t producing time-delayed closed-loop dynamics.Suppose now that protocols (3) and (13) can be designed in terms of the future value, τ units of time, of the agents states, in the form(17)ui(t)=-∑j=1n‍aijxit+τ-xjt+τfor the single integrator case and (18)uit=-∑j=0n‍aijxi1t+τ-xj1t+τhhhhhhhhhh+γxi2t+τ-xj2t+τfor the double integrator agents.It is easy to see that protocols (17) and (18) would solve the delayed consensus problem producing closed-loop systems free of time-delay for the single and double integrator cases. Notice that the convergence properties in the case of system (9)–(17) or (11)–(18) would be equivalent to the ones of the free-delay cases [4, 7–9]. It should be highlighted that protocols (17) and (18) require future values of the agents states that prevent their practical implementation due to their noncausal nature.The main objective of this work is to propose estimated state protocols of the forms (17) and (18) where instead of the actual future states xi(t+τ) and xijt+τ, the use of future estimated states x^i(t+τ) and x^ijt+τ obtained by an adequate prediction observer is considered.


## 4. Prediction-Observer Scheme

As a solution to the delayed consensus problem, for fixed topologies having a spanning tree, a predictor-observer scheme that provides the future estimation of the state of the agents is designed. The scheme has the advantage that the convergence conditions depend on the design parameters and they are not related to the location of the eigenvalues of the Laplacian matrix as it has been previously reported.


### 4.1. Prediction-Observer for Single Integrator Agents

To synthesize the predictor-observer based solution, how the considered agents given in (2) can be expressed in terms of future states is shown first. With this aim, consider the new variables (19)wi(t)=xi(t+τ),for i=1,…,n or equivalently (20)wi(t-τ)=xi(t),producing a new state vector w=[w1,w2,…,wn]T for the group of agents. Taking the time derivative, it is obtained that (21)w˙it-τ=x˙i(t)=ui(t-τ),which by a forward shift of τ units of time produces(22)w˙(t)=u(t).System (22) represents an advanced system, free of delays, for the original agents given by (2).Based on (22), the predictor-observer that will provide the future state of system (2) is defined as(23)w^˙t=u(t)+λew(t-τ),where λ>0 is a design constant parameter, w^∈Rn is the estimation of w(t)=x(t+τ), u(t) is the input signal, and ew(t-τ) is the estimation error, delayed τ units of time defined as(24)ewt-τ=w(t-τ)-w^t-τ,with ewi(t-τ)=wi(t-τ)-w^i(t-τ) for i=1,…,n.Remark 3.
Contrary to a standard Luenberger observer, in the predictor-observer (23), it is not possible to use the signal ew(t) since (25)ew(t)=x(t+τ)-x^(t+τ)that requires future values of the state x(t) which are not available.The convergence of the prediction errorew(t) can be analyzed by considering its time evolution(26)e˙w(t)=-λew(t-τ).It is well known that system (26) is not only asymptotically but also exponentially stable, [35]. The exponential stability property can be formally stated as follows.Definition 4.
The trivial solutionew(t)=0 of the time-delay system (26) is exponentially stable with decay rate α, if there exist constants α>0, β≥1 such that(27)
e
w
(
t
)
≤
β
sup
⁡
-
τ
≤
θ
≤
0
e
w
(
θ
)
e
-
α
t
.The convergence of the predictor-observer (23) is formally stated in the next result.Lemma 5.
Consider a group ofn single integrator agents with a fixed topology having a spanning tree as given in (9) and the predictor-observer defined in (23). Then, for a positive constant λ, signal w^i(t) exponentially converges to the future value xi(t+τ) for i=1,…,n if and only if (28)τ<π2λ.Proof.
The convergence ofw^i(t) can be established by analyzing the stability properties of the prediction error dynamic (26) by noticing that system (26) can be decomposed into the decoupled scalar systems(29)e˙wi(t)=-λewi(t-τ).
The results presented in [36] can be used to show that system (26) will be stable for all τ<τ∗, if and only if,(30)τ∗=π2λ.
In addition, to get an exponential decay rate for system (26), consider the change of variable(31)z(t)=eαtew(t),with z=z1,…,zn∈Rn, α>0. Therefore, (32)z˙t=αeαtew(t)+eαte˙w(t)=αz(t)+eαt(-λew(t-τ))=αz(t)-λeαte-α(t-τ)z(t-τ)yielding(33)z˙(t)=αz(t)-λeατz(t-τ)that produces the decoupled set of differential-difference equations(34)z˙i(t)=αzi(t)-λeατzi(t-τ),for i=1,…,n.
From a direct analysis of the characteristic equation of (34) (see [36, page 119]), it is possible to determine that system (33) will be stable for all τ<τ∗ such that(35)τ∗=arccosα/λeαtλeατ2-α2.
In particular, by assuming that the decay rateα in (31) tends to zero, implying that limα→0eαt=1, it is obtained that(36)limα→0τ∗=limα→0arccosα/λλ2-α2=arccos(0)λ.Since arccos(0)=±2m-1π/2 for m=1,2,…, considering that τ<τ∗ and that τ>0, the following is obtained:(37)limα→0τ∗=π2λ.
Therefore, since system (33) is stable, the solution of (26) is exponentially stable.Remark 6.
The exponential stability of (26) can also be established from [36, Corollary 5.10, page 240]. In this case, the exponential stability is stated for all time-delay τ-, such that(38)τ-<1λwith decay rate σ/τ-, σ>0 that satisfies the transcendence equation (39)-λ+στ+τ-λ2e2σ/1-α=0.In this case, the obtained time-delay bound is more conservative.


### 4.2. Prediction-Observer for Double Integrator Agents

In the case of double integrator agents of the form (11), an equivalent system that describes the future dynamic of the state can be obtained by considering the new variables(40)wi1t=xi1t+τ,wi2t=xi2t+τ,for i=1,…,n. Under these conditions,(41)w˙i1t=x˙i1t+τ=xi2t+τ=wi2t,w˙i2t=x˙i2t+τ=uit;that is,(42)w˙i1t=wi2t,w˙i2t=uit.Defining the estimation errors as(43)ewi1t=wi1t-w^i1t,ewi2t=wi2t-w^i2t,for i=1,…,n, the predictor-observer for system (11) is proposed as(44)w^˙i1t=w^i2t+λi1ewi1t-τ,w^˙i2t=uit+λi0ewi1t-τ.where λi0 and λi1 are positive design constant.Lemma 7.
Consider the group ofn double integrator agents (11) with fixed topology having a spanning tree and the predictor-observer (44). Then the estimation signals w^i1t and w^i2t exponentially converge, respectively, to the future values xi1(t+τ) and xi2t+τ for i=1,…,n if and only if τ<τ∗ such that(45)τ∗=1ωarctanλi1ωλi0,where (46)ω=12λi12+124λi02+λi14for positive design constant gains λi0 and λi1.Proof.
Consider the time derivative of the estimation errors (43):(47)e˙wi1t=w˙i1t-w^˙i1t=wi2t-w^i2t-λi1ewi1t-τ=ewi2t-λi1ewi1t-τ;e˙wi2t=w˙i2t-w^˙i2t=uit-uit-λi0ewit-τ=-λi0ewit-τ,equivalently, (48)e˙wi1t=ewi2t-λi1ewi1t-τ,e˙wi2t=-λi0ewi1t-τ,for i=1,…,n. From (48) it is obtained that (49)e¨wi1t=e˙wi2-λi1e˙wi1t-τ=-λi0ewi1t-τ-λi1e˙wi1t-τ;equivalently,(50)e¨wi1t+λi1e˙wi1t-τ+λi0ewi1t-τ=0.The stability of system (50) can be stated by a direct analysis of its characteristic equation [36]:(51)s2+λi1s+λi0e-τs=0.By considering that the crossing of the roots of (51) with the imaginary axis occurs for s=jω, it is obtained that(52)-ω2+λi0+jλi1ωcos⁡⁡ωτ-jsinωτ=0;that is,(53)-ω2+λi0cos⁡(ωτ)-jλi0sin(ωτ)+jλi1ωcos⁡(ωτ)+λi1ωsin(ωτ)=0;therefore,(54a)-ω2+λi0cos⁡(ωτ)+λi1ωsin(ωτ)=0(54b)-λi0sin(ωτ)+λi1ωcos⁡(ωτ)=0.From (54b), (55)τ=1ωarctanλi1ωλi0which substituted in (54a) produces(56)λi0cos⁡arctanλi1ωλi0+λi1ωsinarctanλi1ωλi0=ω2,from where(57)λi011+λi1ω/λi02+λi1ωλi1ω/λi01+λi1ω/λi02=ω2.Equivalently,(58)λi02+λi12ω2λi0ω2=1+λi1ωλi02,that after some simplifications yields(59)ω6+λi02-λi14λi12ω4-2λi02ω2-λi04λi12=0.To get a positive solution of (59), consider the auxiliary variable(60)ζ=ω2.Then (59) is rewritten in the form (61)ζ+λi02λi12ζ2-λi12ζ-λi02=0with the solutions(62)ζ1=-λi02λi12,ζ2=12λi12+124λi02+λi14,ζ3=12λi12-124λi02+λi14.Since λi0 and λi1 are positive constants, it is easy to see that ζ2 is the only positive solution. Equations (55) and w=ζ2 show the convergence of the estimation signals.
In order to get an exponential decay rate for system (48), consider its equivalent description(63)e˙wi(t)=Aewi(t)+Adewi(t-τ),where ewi=ewi1,ewi2T,(64)A=0100,Ad=-λi10-λi00,and the change of variable(65)ξit=eαtewit,α>0.Then, it is possible to write(66)ewit=e-αtξit,ewit-τ=e-αt-τξit-τ,and therefore (67)ξ˙it=αeαtewit+eαte˙wit=αeαtewit+eαtAewit+Adewit-τ=A+αIeαtewit+eαteαt-τe-αt-τAdewit-τ=A+αIeαtewit+eατAdeαt-τewit-τ,obtaining(68)ξ˙it=A+αIξit+eατAdξit-τ.Notice that if ξit is stable, then, by (65), ewit will be exponentially stable with decay rate α. Since (63) is stable, then, by means of a continuity argument, the stability of (68) can always be established for a sufficiently small constant α.


## 4.1. Prediction-Observer for Single Integrator Agents

To synthesize the predictor-observer based solution, how the considered agents given in (2) can be expressed in terms of future states is shown first. With this aim, consider the new variables (19)wi(t)=xi(t+τ),for i=1,…,n or equivalently (20)wi(t-τ)=xi(t),producing a new state vector w=[w1,w2,…,wn]T for the group of agents. Taking the time derivative, it is obtained that (21)w˙it-τ=x˙i(t)=ui(t-τ),which by a forward shift of τ units of time produces(22)w˙(t)=u(t).System (22) represents an advanced system, free of delays, for the original agents given by (2).Based on (22), the predictor-observer that will provide the future state of system (2) is defined as(23)w^˙t=u(t)+λew(t-τ),where λ>0 is a design constant parameter, w^∈Rn is the estimation of w(t)=x(t+τ), u(t) is the input signal, and ew(t-τ) is the estimation error, delayed τ units of time defined as(24)ewt-τ=w(t-τ)-w^t-τ,with ewi(t-τ)=wi(t-τ)-w^i(t-τ) for i=1,…,n.Remark 3.
Contrary to a standard Luenberger observer, in the predictor-observer (23), it is not possible to use the signal ew(t) since (25)ew(t)=x(t+τ)-x^(t+τ)that requires future values of the state x(t) which are not available.The convergence of the prediction errorew(t) can be analyzed by considering its time evolution(26)e˙w(t)=-λew(t-τ).It is well known that system (26) is not only asymptotically but also exponentially stable, [35]. The exponential stability property can be formally stated as follows.Definition 4.
The trivial solutionew(t)=0 of the time-delay system (26) is exponentially stable with decay rate α, if there exist constants α>0, β≥1 such that(27)
e
w
(
t
)
≤
β
sup
⁡
-
τ
≤
θ
≤
0
e
w
(
θ
)
e
-
α
t
.The convergence of the predictor-observer (23) is formally stated in the next result.Lemma 5.
Consider a group ofn single integrator agents with a fixed topology having a spanning tree as given in (9) and the predictor-observer defined in (23). Then, for a positive constant λ, signal w^i(t) exponentially converges to the future value xi(t+τ) for i=1,…,n if and only if (28)τ<π2λ.Proof.
The convergence ofw^i(t) can be established by analyzing the stability properties of the prediction error dynamic (26) by noticing that system (26) can be decomposed into the decoupled scalar systems(29)e˙wi(t)=-λewi(t-τ).
The results presented in [36] can be used to show that system (26) will be stable for all τ<τ∗, if and only if,(30)τ∗=π2λ.
In addition, to get an exponential decay rate for system (26), consider the change of variable(31)z(t)=eαtew(t),with z=z1,…,zn∈Rn, α>0. Therefore, (32)z˙t=αeαtew(t)+eαte˙w(t)=αz(t)+eαt(-λew(t-τ))=αz(t)-λeαte-α(t-τ)z(t-τ)yielding(33)z˙(t)=αz(t)-λeατz(t-τ)that produces the decoupled set of differential-difference equations(34)z˙i(t)=αzi(t)-λeατzi(t-τ),for i=1,…,n.
From a direct analysis of the characteristic equation of (34) (see [36, page 119]), it is possible to determine that system (33) will be stable for all τ<τ∗ such that(35)τ∗=arccosα/λeαtλeατ2-α2.
In particular, by assuming that the decay rateα in (31) tends to zero, implying that limα→0eαt=1, it is obtained that(36)limα→0τ∗=limα→0arccosα/λλ2-α2=arccos(0)λ.Since arccos(0)=±2m-1π/2 for m=1,2,…, considering that τ<τ∗ and that τ>0, the following is obtained:(37)limα→0τ∗=π2λ.
Therefore, since system (33) is stable, the solution of (26) is exponentially stable.Remark 6.
The exponential stability of (26) can also be established from [36, Corollary 5.10, page 240]. In this case, the exponential stability is stated for all time-delay τ-, such that(38)τ-<1λwith decay rate σ/τ-, σ>0 that satisfies the transcendence equation (39)-λ+στ+τ-λ2e2σ/1-α=0.In this case, the obtained time-delay bound is more conservative.


## 4.2. Prediction-Observer for Double Integrator Agents

In the case of double integrator agents of the form (11), an equivalent system that describes the future dynamic of the state can be obtained by considering the new variables(40)wi1t=xi1t+τ,wi2t=xi2t+τ,for i=1,…,n. Under these conditions,(41)w˙i1t=x˙i1t+τ=xi2t+τ=wi2t,w˙i2t=x˙i2t+τ=uit;that is,(42)w˙i1t=wi2t,w˙i2t=uit.Defining the estimation errors as(43)ewi1t=wi1t-w^i1t,ewi2t=wi2t-w^i2t,for i=1,…,n, the predictor-observer for system (11) is proposed as(44)w^˙i1t=w^i2t+λi1ewi1t-τ,w^˙i2t=uit+λi0ewi1t-τ.where λi0 and λi1 are positive design constant.Lemma 7.
Consider the group ofn double integrator agents (11) with fixed topology having a spanning tree and the predictor-observer (44). Then the estimation signals w^i1t and w^i2t exponentially converge, respectively, to the future values xi1(t+τ) and xi2t+τ for i=1,…,n if and only if τ<τ∗ such that(45)τ∗=1ωarctanλi1ωλi0,where (46)ω=12λi12+124λi02+λi14for positive design constant gains λi0 and λi1.Proof.
Consider the time derivative of the estimation errors (43):(47)e˙wi1t=w˙i1t-w^˙i1t=wi2t-w^i2t-λi1ewi1t-τ=ewi2t-λi1ewi1t-τ;e˙wi2t=w˙i2t-w^˙i2t=uit-uit-λi0ewit-τ=-λi0ewit-τ,equivalently, (48)e˙wi1t=ewi2t-λi1ewi1t-τ,e˙wi2t=-λi0ewi1t-τ,for i=1,…,n. From (48) it is obtained that (49)e¨wi1t=e˙wi2-λi1e˙wi1t-τ=-λi0ewi1t-τ-λi1e˙wi1t-τ;equivalently,(50)e¨wi1t+λi1e˙wi1t-τ+λi0ewi1t-τ=0.The stability of system (50) can be stated by a direct analysis of its characteristic equation [36]:(51)s2+λi1s+λi0e-τs=0.By considering that the crossing of the roots of (51) with the imaginary axis occurs for s=jω, it is obtained that(52)-ω2+λi0+jλi1ωcos⁡⁡ωτ-jsinωτ=0;that is,(53)-ω2+λi0cos⁡(ωτ)-jλi0sin(ωτ)+jλi1ωcos⁡(ωτ)+λi1ωsin(ωτ)=0;therefore,(54a)-ω2+λi0cos⁡(ωτ)+λi1ωsin(ωτ)=0(54b)-λi0sin(ωτ)+λi1ωcos⁡(ωτ)=0.From (54b), (55)τ=1ωarctanλi1ωλi0which substituted in (54a) produces(56)λi0cos⁡arctanλi1ωλi0+λi1ωsinarctanλi1ωλi0=ω2,from where(57)λi011+λi1ω/λi02+λi1ωλi1ω/λi01+λi1ω/λi02=ω2.Equivalently,(58)λi02+λi12ω2λi0ω2=1+λi1ωλi02,that after some simplifications yields(59)ω6+λi02-λi14λi12ω4-2λi02ω2-λi04λi12=0.To get a positive solution of (59), consider the auxiliary variable(60)ζ=ω2.Then (59) is rewritten in the form (61)ζ+λi02λi12ζ2-λi12ζ-λi02=0with the solutions(62)ζ1=-λi02λi12,ζ2=12λi12+124λi02+λi14,ζ3=12λi12-124λi02+λi14.Since λi0 and λi1 are positive constants, it is easy to see that ζ2 is the only positive solution. Equations (55) and w=ζ2 show the convergence of the estimation signals.
In order to get an exponential decay rate for system (48), consider its equivalent description(63)e˙wi(t)=Aewi(t)+Adewi(t-τ),where ewi=ewi1,ewi2T,(64)A=0100,Ad=-λi10-λi00,and the change of variable(65)ξit=eαtewit,α>0.Then, it is possible to write(66)ewit=e-αtξit,ewit-τ=e-αt-τξit-τ,and therefore (67)ξ˙it=αeαtewit+eαte˙wit=αeαtewit+eαtAewit+Adewit-τ=A+αIeαtewit+eαteαt-τe-αt-τAdewit-τ=A+αIeαtewit+eατAdeαt-τewit-τ,obtaining(68)ξ˙it=A+αIξit+eατAdξit-τ.Notice that if ξit is stable, then, by (65), ewit will be exponentially stable with decay rate α. Since (63) is stable, then, by means of a continuity argument, the stability of (68) can always be established for a sufficiently small constant α.


## 5. Solution to the Consensus Problem

### 5.1. Single Integrator Agents Case

The solution for the consensus problem proposed in [9] for system (9), given in (3), can be redefined by considering the estimated future states, resulting in (69)u(t)=-Lw^(t)=-Lx^(t+τ),producing the closed-loop (9)–(69) as(70)x˙t=-Lx^t,w^˙t=-Lw^(t)+λew(t-τ).Therefore, by considering the estimation error (24) (71)ew(t-τ)=w(t-τ)-w^(t-τ)=x(t)-x^(t),system (70) can be rewritten as (72a)x˙t=-Lxt+Lewt-τ,(72b)e˙w(t)=-λew(t-τ),where, as defined previously, x=x1,…,xnT, ew=ew1,…,ewnT, λ>0, and L∈Rn×n is the Laplacian matrix related to the topology of the agents.By considering a nonsingular matrixP=[v1,…,vn]∈Rn×n defined by the generalized eigenvectors vi of matrix L, it is possible to define the change of coordinates (73)x=Pχ,χ=P-1x,producing a new representation for (72a) in the form(74)χ˙=-Jχ+P-1Lew(t-τ),where J=P-1LP is the Jordan canonical form of matrix L.The linear nature of (74) allows writing its solution as(75)χ(t)=e-Jtχ(0)+∫0t‍e-J(t-s)P-1Lew(s-τ)ds.From the properties ofL given in Section 2, the generalized eigenvector associated with the zero eigenvalue satisfies v1=1n; therefore, (76)J=P-1LP=00000B10000⋱0000Br=000J-,where Bi∈Rmi×mi correspond to Jordan blocks related to the independent eigenvalues λi≠0, with multiplicity mi, i=1,…,r, of matrix L. From the structure of J in (76), 1+m1+⋯+mr=n and J-∈Rn-1×n-1 [37].To get the solution of the consensus problem, note that, in (75), JP-1=P-1L and therefore(77)χt=100e-J-tχ0+∫0t‍100e-J-t-s000J-P-1ews-τds;equivalently,(78)χt=100e-J-tχ0+0∫0t‍e-J-t-sJ-P-1ews-τds.Equation (78) corresponds to the solution of the subsystems(79a)χ¯˙1t=0,(79b)χ¯˙2t=J¯χ¯2t+ftwith χ-1=χ1, χ-2=χ2⋯χnT, and f(t)=J-P-1ew(t-τ). By standard properties of linear systems, notice that since limt→∞f(t)=0, then, when t→∞, the solution of the nonautonomous system (79b) corresponds to its homogeneous solution, and therefore it is possible to assure that(80)limt→∞∫0t‍e-J(t-s)P-1Lew(s-τ)ds=0.It is clear now fromx(t)=Pχ(t) and (78) that (81)limt→∞x(t)=P1000(n-1)×(n-1)χ(0)=Pχ1(0)0⋮0.Since the first column of P is given by 1n, (82)limt→∞xt=χ10⋮χ10=P1i-1x0⋮P1i-1x0,where P1i-1 correspond to the first row of matrix P-1.Additionally, from (82), it is shown that states xi(t) for i=1,…,n converge to the same value(83)xi(∞)=χi(0)=P1i-1x(0)that depends on the agents initial conditions.The preceding developments allow formally stating the following result.Theorem 8.
Consider a group ofn single integrator agents with a fixed topology having a spanning tree as given in (9). Protocol (69) based on predicted states given by (23) solves the delayed consensus problem if and only if τ<π/2λ. Moreover, xi(t)→P1i-1x(0).Remark 9.
In fact it is not necessary that functionf(t) in (79b) tends exponentially to zero; it suffices its asymptotic convergence. The exponential property is important in order to improve the rate of convergence to the agreement of the agents.


### 5.2. Double Integrator Agents Case

From the definition of the advance variables (40) and the predictor-observer (44), it is possible to see that the future estimation of the state x^ijt+τ for i=1,…,n and j=1,2 is defined as(84)x^ijt+τ=w^ijt.In this case, the noncausal feedback (18) can be modified as (85)uit=-∑j=1n‍aijw^i1t-w^j1thhhhhhhhhh+γw^i2t-w^j2t,which in compact form results(86)ut=-Lw^1t-γLw^2t,where w^j=w^1j,…,w^njT, j=1,2, and L∈Rn×n is the Laplacian generated by the network topology.Feedback (86), in closed-loop with system (12), produces(87)x˙1t=x2t,x˙2t=-Lw^1t-τ-γLw^2t-τ.By considering definition (84), (43) can be expressed as ewj(t-τ)=xj(t)-x^j(t), for j=1,2, turning (87) in(88)x˙1t=x2t,x˙2t=-Lx1t-γLx2t+Lew1t-τ-γLew2t-τ.The overall dynamic of the closed-loop (12)–(86) is obtained from (88) and (48), resulting in(89)x˙1=x2,x˙2=-Lx1-γLx2+Lew1t-τ-γLew2t-τ,e˙w1=ew2-λ1ew1t-τ,e˙w2=-λ0ew1t-τ.Notice that, regarding (40), system (89) can be rewritten in terms of the future variables, yielding the representation(90a)w˙1=w2w˙2=-Lw1-γLw2+Lew1+γLew2(90b)e˙w1=ew2-λ1ew1t-τe˙w2=-λ0ew1t-τ,where wj=w1j,…,wnjT, for j=1,2.Taking into account from (68) that subsystem (90b) is exponentially stable, it is clear that (91)Lw1,w2=0,0is an equilibrium point of the subsystem (90a). Notice that Lw1=0 is satisfied for w11=w21=⋯=wn1, since the sum of the elements of each row of L is always null. Notice also that the convergence of wi1(t)=xi1(t+τ) and wi2(t)=xi2(t+τ) implies the convergence of the states xi1(t) and xi2(t) due to the finite value of τ.


### 5.3. Position and Velocity Convergence

The dynamic of system (90a) can be written alternatively by considering the transformation (92)w1w2=P00Pφ1φ2with matrix P as in (76) yielding(93)φ˙1=φ2φ˙2=-P-1LPφ1-γP-1LPφ2+P-1Lew1+γP-1Lew2.SinceP-1LP=J, with J as the Jordan form of L, then (93) can be rewritten as(94)φ˙=ΓJφ+BLewfor φ=φ1,φ2T with φ1=φ11…φ1nT, φ1=φ21…φ2nT, and (95)ΓJ=0I-J-γJ,BL=00P-1LγP-1L.The characteristic polynomial of the autonomous system(96)φ˙=ΓJφis obtained as(97)pφs=det⁡sI-ΓJ=det⁡s2I+γJs+J=det⁡Is2+1+γsJ=det⁡P-1Is2+1+γsLP.By considering the complex Jordan formJ with eigenvalues μi, i=1,…,n, it is obtained that(98)pφs=det⁡sI-ΓJ=∏i=1ns2+1+γsμifrom where it is clear that each eigenvalue of L produces two eigenvalues of ΓJ [7].Lemma 10.
Suppose thatμi≠0 is an eigenvalue of the Laplacian matrix L associated with a communication topology having a spanning tree of a group of n double integrators agents. Then, every μi will produce two eigenvalues μJi of ΓJ with ReμJi<0 if and only if(99)γ>max⁡μi≠01μiImμiReμi.Proof.
From (98), the eigenvalues of ΓJ can be obtained from the equation(100)s2+γμis+μi=0for i=1,…,n-1, in the form(101)s1,2=12-γμi±γ2μi2-4μi.
By considering thatμi=ai+jbi with j2=-1, it is obtained that(102)2s1,2=-γai-jγbi±γ2ai2-bi2-4ai+j2γ2aibi-4bi=-γai-jγbi±12γ2ai2-bi2-4ai2+2γ2aibi-4bi2+γ2ai2-bi2-4aihhhhh+jsgn2γ2aibi-4bi2hhhhhh×γ2ai2-bi2-4ai2+2γ2aibi-4bi2-γ2ai2-bi2+4ai.
Therefore, to assureReμJi<0, the following inequality is considered:(103)-γai+12γ2ai2-bi2-4ai2+2γ2aibi-4bi2+γ2ai2-bi2-4ai<0that produces(104)2γ2ai2-γ2ai2-bi2-4ai2>γ2ai2-bi2-4ai2+2γ2aibi-4bi2.
After some manipulations, it is obtained that(105)γ2>bi2ai1ai2+bi2.
This proves the lemma.Remark 11.
It should be noticed that Lemma10 can be directly applied to the consensus for double integrator agents free of delay. In particular, Lemma 10 improves the bound reported on Lemma 4.4 and Theorem  4.2 in [7] providing necessary and sufficient conditions instead of only sufficient ones.Based on Lemma10, the solution to the consensus protocol for double integrator agents is stated as follows.Theorem 12.
Consider a group ofn double integrator agents with a fixed information topology having a spanning tree. Protocol (85) solves the consensus problem associated to double integrator agents (11), if and only if (106)γ>max⁡μi≠01μiImμiReμi.In particular x1∞=1nT1i-1+tT2i-1x0 and x2(∞)=1nT2i-1x0 for T1i-1 and T2i-1 defined, respectively, by the first and second row of the inverse of matrix T=t1,…,t2n, formed by the generalized eigenvectors ti of ΓJ.Proof.
Taking into account the structure of the matrixJ given in (76), (93) can be rewritten as(107a)φ˙11=φ21,φ˙21=0,(107b)φ˙1b=φ2b,φ˙2b=-J-φ1b-γJ-φ2b+J-P-1ew1+γJ-P-1ew2,where φ11,φ21∈R and φ1b,φ2b∈Rn-1. From Lemma 10, subsystem (107b) is exponentially stable with an exogenous input f(t)=J-P-1ew1+γJ-P-1ew2. Therefore, since limt→∞f(t)=0, the state convergence of system (107a) and (107b) is obtained by considering the response of the autonomous system (108)φ˙11φ˙21φ˙1bφ˙2b=010002×(2n-2)0(2n-2)×20(n-1)×(n-1)I(n-1)×(n-1)-J--γJ-φ11φ21φ1bφ2b=Aφφ11φ21φ1bφ2b.
Define now a transformation matrixT formed by the generalized eigenvectors t1,t2,…,t2n of ΓJ; T=t1,…,t2n. Notice that the first diagonal block of Aφ is already the Jordan block associated with the pair of zero eigenvalues μ1=μ2=0 of Aφ. By the uniqueness of the Jordan canonical form, up to the order of the eigenvalues, the corresponding eigenvectors are given by the first and n+1th columns of the matrix considered in transformation (92) given as [7](109)t1=1n0n,t2=0n1n.
The remaining eigenvectors,t3,…,t2n, will transform the second block of Aφ to the Jordan form. Therefore, by defining the transformation w=Tψ, the equivalent autonomous system is obtained:(110)ψ˙=JTψfor ψ=ψ1,ψ2T with ψ1=ψ11,ψ12T, ψ2=ψ21,…,ψ2(2n-2)T, and(111)JT=T-1ΓT.
In this case,JT takes the form(112)JT=010002×(2n-2)0(2n-2)×2J-Twith J-T∈R2n-2×2n-2 in a Jordan block. The time solution of (110) can be easily obtained as(113)ψt=eJTtψ0from where the evolution of the future state can be obtained in the form (114)wt=TeJTtψ0.
By considering (112), the future states converge according to(115)limt→∞wt=T1t0100eJ-Ttψ0.
Notice that since all the eigenvalues associated withJ-T are negative, the following is obtained:(116)limt→∞wt=Tψ110+tψ210ψ2100⋮0,in addition, the general structure of T allows writing (117)w∞=1n0n∗0n1n∗ψ110+tψ210ψ2100⋮0=1nψ110+tψ2101nψ210.
Sinceψ=T-1w, (118)w1∞=1nT1i-1+tT2i-1w0,w2∞=1nT2i-1w0,where T1i-1 and T2i-1 represent, respectively, the first and second row of matrix T-1.
To complete the proof, notice that, due to relation (40), a time shift of τ units of time provides(119)x1∞=1nT1i-1+tT2i-1x0,x2(∞)=1nT2i-1x0.The computation of the final values given in (119) can be simplified by obtaining T-1 by means of a set on n independent left eigenvectors hi,  i=1,…,2n of T. It has been proved in [7] that h1=pT0nTT and h2=0nTpTT are the left eigenvectors associated with the zero eigenvalue with p being a vector that satisfies pTL=0 and PT1n=1 for P as in (73). A direct computation of hi and h2 can be done by considering pTL=pTLP=0 which implies that(120)pTPJ=0.SinceP=[v1,…,vn] with v1=1n, then pTv1=1. By the structure of matrix J and condition (120), it is obtained that pTvi=0 for i=2,…,n. It is clear now that pT correspond to the first row of matrix P-1 denoted in Theorem 8 as P1i-1. Direct computations show that(121)x1∞=1nP1i-1x1(0)+tx2(0),x2(∞)=1nP1i-1x20.Therefore, the knowledge of matricesL and P gives the information to compute the value of the agreement.


## 5.1. Single Integrator Agents Case

The solution for the consensus problem proposed in [9] for system (9), given in (3), can be redefined by considering the estimated future states, resulting in (69)u(t)=-Lw^(t)=-Lx^(t+τ),producing the closed-loop (9)–(69) as(70)x˙t=-Lx^t,w^˙t=-Lw^(t)+λew(t-τ).Therefore, by considering the estimation error (24) (71)ew(t-τ)=w(t-τ)-w^(t-τ)=x(t)-x^(t),system (70) can be rewritten as (72a)x˙t=-Lxt+Lewt-τ,(72b)e˙w(t)=-λew(t-τ),where, as defined previously, x=x1,…,xnT, ew=ew1,…,ewnT, λ>0, and L∈Rn×n is the Laplacian matrix related to the topology of the agents.By considering a nonsingular matrixP=[v1,…,vn]∈Rn×n defined by the generalized eigenvectors vi of matrix L, it is possible to define the change of coordinates (73)x=Pχ,χ=P-1x,producing a new representation for (72a) in the form(74)χ˙=-Jχ+P-1Lew(t-τ),where J=P-1LP is the Jordan canonical form of matrix L.The linear nature of (74) allows writing its solution as(75)χ(t)=e-Jtχ(0)+∫0t‍e-J(t-s)P-1Lew(s-τ)ds.From the properties ofL given in Section 2, the generalized eigenvector associated with the zero eigenvalue satisfies v1=1n; therefore, (76)J=P-1LP=00000B10000⋱0000Br=000J-,where Bi∈Rmi×mi correspond to Jordan blocks related to the independent eigenvalues λi≠0, with multiplicity mi, i=1,…,r, of matrix L. From the structure of J in (76), 1+m1+⋯+mr=n and J-∈Rn-1×n-1 [37].To get the solution of the consensus problem, note that, in (75), JP-1=P-1L and therefore(77)χt=100e-J-tχ0+∫0t‍100e-J-t-s000J-P-1ews-τds;equivalently,(78)χt=100e-J-tχ0+0∫0t‍e-J-t-sJ-P-1ews-τds.Equation (78) corresponds to the solution of the subsystems(79a)χ¯˙1t=0,(79b)χ¯˙2t=J¯χ¯2t+ftwith χ-1=χ1, χ-2=χ2⋯χnT, and f(t)=J-P-1ew(t-τ). By standard properties of linear systems, notice that since limt→∞f(t)=0, then, when t→∞, the solution of the nonautonomous system (79b) corresponds to its homogeneous solution, and therefore it is possible to assure that(80)limt→∞∫0t‍e-J(t-s)P-1Lew(s-τ)ds=0.It is clear now fromx(t)=Pχ(t) and (78) that (81)limt→∞x(t)=P1000(n-1)×(n-1)χ(0)=Pχ1(0)0⋮0.Since the first column of P is given by 1n, (82)limt→∞xt=χ10⋮χ10=P1i-1x0⋮P1i-1x0,where P1i-1 correspond to the first row of matrix P-1.Additionally, from (82), it is shown that states xi(t) for i=1,…,n converge to the same value(83)xi(∞)=χi(0)=P1i-1x(0)that depends on the agents initial conditions.The preceding developments allow formally stating the following result.Theorem 8.
Consider a group ofn single integrator agents with a fixed topology having a spanning tree as given in (9). Protocol (69) based on predicted states given by (23) solves the delayed consensus problem if and only if τ<π/2λ. Moreover, xi(t)→P1i-1x(0).Remark 9.
In fact it is not necessary that functionf(t) in (79b) tends exponentially to zero; it suffices its asymptotic convergence. The exponential property is important in order to improve the rate of convergence to the agreement of the agents.


## 5.2. Double Integrator Agents Case

From the definition of the advance variables (40) and the predictor-observer (44), it is possible to see that the future estimation of the state x^ijt+τ for i=1,…,n and j=1,2 is defined as(84)x^ijt+τ=w^ijt.In this case, the noncausal feedback (18) can be modified as (85)uit=-∑j=1n‍aijw^i1t-w^j1thhhhhhhhhh+γw^i2t-w^j2t,which in compact form results(86)ut=-Lw^1t-γLw^2t,where w^j=w^1j,…,w^njT, j=1,2, and L∈Rn×n is the Laplacian generated by the network topology.Feedback (86), in closed-loop with system (12), produces(87)x˙1t=x2t,x˙2t=-Lw^1t-τ-γLw^2t-τ.By considering definition (84), (43) can be expressed as ewj(t-τ)=xj(t)-x^j(t), for j=1,2, turning (87) in(88)x˙1t=x2t,x˙2t=-Lx1t-γLx2t+Lew1t-τ-γLew2t-τ.The overall dynamic of the closed-loop (12)–(86) is obtained from (88) and (48), resulting in(89)x˙1=x2,x˙2=-Lx1-γLx2+Lew1t-τ-γLew2t-τ,e˙w1=ew2-λ1ew1t-τ,e˙w2=-λ0ew1t-τ.Notice that, regarding (40), system (89) can be rewritten in terms of the future variables, yielding the representation(90a)w˙1=w2w˙2=-Lw1-γLw2+Lew1+γLew2(90b)e˙w1=ew2-λ1ew1t-τe˙w2=-λ0ew1t-τ,where wj=w1j,…,wnjT, for j=1,2.Taking into account from (68) that subsystem (90b) is exponentially stable, it is clear that (91)Lw1,w2=0,0is an equilibrium point of the subsystem (90a). Notice that Lw1=0 is satisfied for w11=w21=⋯=wn1, since the sum of the elements of each row of L is always null. Notice also that the convergence of wi1(t)=xi1(t+τ) and wi2(t)=xi2(t+τ) implies the convergence of the states xi1(t) and xi2(t) due to the finite value of τ.


## 5.3. Position and Velocity Convergence

The dynamic of system (90a) can be written alternatively by considering the transformation (92)w1w2=P00Pφ1φ2with matrix P as in (76) yielding(93)φ˙1=φ2φ˙2=-P-1LPφ1-γP-1LPφ2+P-1Lew1+γP-1Lew2.SinceP-1LP=J, with J as the Jordan form of L, then (93) can be rewritten as(94)φ˙=ΓJφ+BLewfor φ=φ1,φ2T with φ1=φ11…φ1nT, φ1=φ21…φ2nT, and (95)ΓJ=0I-J-γJ,BL=00P-1LγP-1L.The characteristic polynomial of the autonomous system(96)φ˙=ΓJφis obtained as(97)pφs=det⁡sI-ΓJ=det⁡s2I+γJs+J=det⁡Is2+1+γsJ=det⁡P-1Is2+1+γsLP.By considering the complex Jordan formJ with eigenvalues μi, i=1,…,n, it is obtained that(98)pφs=det⁡sI-ΓJ=∏i=1ns2+1+γsμifrom where it is clear that each eigenvalue of L produces two eigenvalues of ΓJ [7].Lemma 10.
Suppose thatμi≠0 is an eigenvalue of the Laplacian matrix L associated with a communication topology having a spanning tree of a group of n double integrators agents. Then, every μi will produce two eigenvalues μJi of ΓJ with ReμJi<0 if and only if(99)γ>max⁡μi≠01μiImμiReμi.Proof.
From (98), the eigenvalues of ΓJ can be obtained from the equation(100)s2+γμis+μi=0for i=1,…,n-1, in the form(101)s1,2=12-γμi±γ2μi2-4μi.
By considering thatμi=ai+jbi with j2=-1, it is obtained that(102)2s1,2=-γai-jγbi±γ2ai2-bi2-4ai+j2γ2aibi-4bi=-γai-jγbi±12γ2ai2-bi2-4ai2+2γ2aibi-4bi2+γ2ai2-bi2-4aihhhhh+jsgn2γ2aibi-4bi2hhhhhh×γ2ai2-bi2-4ai2+2γ2aibi-4bi2-γ2ai2-bi2+4ai.
Therefore, to assureReμJi<0, the following inequality is considered:(103)-γai+12γ2ai2-bi2-4ai2+2γ2aibi-4bi2+γ2ai2-bi2-4ai<0that produces(104)2γ2ai2-γ2ai2-bi2-4ai2>γ2ai2-bi2-4ai2+2γ2aibi-4bi2.
After some manipulations, it is obtained that(105)γ2>bi2ai1ai2+bi2.
This proves the lemma.Remark 11.
It should be noticed that Lemma10 can be directly applied to the consensus for double integrator agents free of delay. In particular, Lemma 10 improves the bound reported on Lemma 4.4 and Theorem  4.2 in [7] providing necessary and sufficient conditions instead of only sufficient ones.Based on Lemma10, the solution to the consensus protocol for double integrator agents is stated as follows.Theorem 12.
Consider a group ofn double integrator agents with a fixed information topology having a spanning tree. Protocol (85) solves the consensus problem associated to double integrator agents (11), if and only if (106)γ>max⁡μi≠01μiImμiReμi.In particular x1∞=1nT1i-1+tT2i-1x0 and x2(∞)=1nT2i-1x0 for T1i-1 and T2i-1 defined, respectively, by the first and second row of the inverse of matrix T=t1,…,t2n, formed by the generalized eigenvectors ti of ΓJ.Proof.
Taking into account the structure of the matrixJ given in (76), (93) can be rewritten as(107a)φ˙11=φ21,φ˙21=0,(107b)φ˙1b=φ2b,φ˙2b=-J-φ1b-γJ-φ2b+J-P-1ew1+γJ-P-1ew2,where φ11,φ21∈R and φ1b,φ2b∈Rn-1. From Lemma 10, subsystem (107b) is exponentially stable with an exogenous input f(t)=J-P-1ew1+γJ-P-1ew2. Therefore, since limt→∞f(t)=0, the state convergence of system (107a) and (107b) is obtained by considering the response of the autonomous system (108)φ˙11φ˙21φ˙1bφ˙2b=010002×(2n-2)0(2n-2)×20(n-1)×(n-1)I(n-1)×(n-1)-J--γJ-φ11φ21φ1bφ2b=Aφφ11φ21φ1bφ2b.
Define now a transformation matrixT formed by the generalized eigenvectors t1,t2,…,t2n of ΓJ; T=t1,…,t2n. Notice that the first diagonal block of Aφ is already the Jordan block associated with the pair of zero eigenvalues μ1=μ2=0 of Aφ. By the uniqueness of the Jordan canonical form, up to the order of the eigenvalues, the corresponding eigenvectors are given by the first and n+1th columns of the matrix considered in transformation (92) given as [7](109)t1=1n0n,t2=0n1n.
The remaining eigenvectors,t3,…,t2n, will transform the second block of Aφ to the Jordan form. Therefore, by defining the transformation w=Tψ, the equivalent autonomous system is obtained:(110)ψ˙=JTψfor ψ=ψ1,ψ2T with ψ1=ψ11,ψ12T, ψ2=ψ21,…,ψ2(2n-2)T, and(111)JT=T-1ΓT.
In this case,JT takes the form(112)JT=010002×(2n-2)0(2n-2)×2J-Twith J-T∈R2n-2×2n-2 in a Jordan block. The time solution of (110) can be easily obtained as(113)ψt=eJTtψ0from where the evolution of the future state can be obtained in the form (114)wt=TeJTtψ0.
By considering (112), the future states converge according to(115)limt→∞wt=T1t0100eJ-Ttψ0.
Notice that since all the eigenvalues associated withJ-T are negative, the following is obtained:(116)limt→∞wt=Tψ110+tψ210ψ2100⋮0,in addition, the general structure of T allows writing (117)w∞=1n0n∗0n1n∗ψ110+tψ210ψ2100⋮0=1nψ110+tψ2101nψ210.
Sinceψ=T-1w, (118)w1∞=1nT1i-1+tT2i-1w0,w2∞=1nT2i-1w0,where T1i-1 and T2i-1 represent, respectively, the first and second row of matrix T-1.
To complete the proof, notice that, due to relation (40), a time shift of τ units of time provides(119)x1∞=1nT1i-1+tT2i-1x0,x2(∞)=1nT2i-1x0.The computation of the final values given in (119) can be simplified by obtaining T-1 by means of a set on n independent left eigenvectors hi,  i=1,…,2n of T. It has been proved in [7] that h1=pT0nTT and h2=0nTpTT are the left eigenvectors associated with the zero eigenvalue with p being a vector that satisfies pTL=0 and PT1n=1 for P as in (73). A direct computation of hi and h2 can be done by considering pTL=pTLP=0 which implies that(120)pTPJ=0.SinceP=[v1,…,vn] with v1=1n, then pTv1=1. By the structure of matrix J and condition (120), it is obtained that pTvi=0 for i=2,…,n. It is clear now that pT correspond to the first row of matrix P-1 denoted in Theorem 8 as P1i-1. Direct computations show that(121)x1∞=1nP1i-1x1(0)+tx2(0),x2(∞)=1nP1i-1x20.Therefore, the knowledge of matricesL and P gives the information to compute the value of the agreement.


## 6. Numeric Evaluation

To show the effectiveness of the prediction-based solution of the delayed consensus problem, two numerical problems will be considered.


### 6.1. Example 1

To compare the proposed solution with a previous result, this example is inspired from [9], where three agents with the topology depicted in Figure 1 are considered. The associated Laplacian matrix takes the form(122)L=10-1-110-1-12,where λmax⁡(L)=2. Following the control strategy presented in [9], the delayed consensus problem can be solved for τ<π/2λmax⁡(L)=0.785 s.Figure 1
Three single integrator agents topology.When applying the prediction-based strategy proposed in this work, it can be shown that by considering the design parameterλ=1.0472 a maximum time-delay τ∗=1.5 s is obtained. For this experiment, τ=0.785 and the initial conditions x(0)=[-4,1,3]T are considered. The convergence to zero of the observation error ew(t-τ)=x(t)-x^(t) is shown in Figure 2(a) while the consensus of the agents state is shown in Figure 2(b). The evolution of the control signals used on the experiment is depicted in Figure 2(c).Figure 2
Single integrator case,τ=0.785 s.
(a)
Observation-prediction erroreω(t-τ)
(b)
System outputx(t)
(c)
Control signalu(t)In a more challenging situation, where the time-delay is set asτ=1.2 s, Figure 3(a) shows the time evolution of the delayed prediction error. Figure 3(b) depicts the evolution of the states and the corresponding control signals are shown in Figure 3(c). Notice how the rate of convergence for this case is slower than the previous one due to the fact that the delay is near the maximum allowable delay time.Figure 3
Single integrator case,τ=1.2 s.
(a)
Observation-prediction erroreω(t-τ)
(b)
System outputx(t)
(c)
Control signalu(t)To determine the convergence value of the agents’ state, notice that matrixP given in (73) takes the form (123)P=1-12-14112-1411234.Therefore, by considering (82), xi(t)→P1i-1x(0); that is,(124)P1i-1x(0)=0.50.250.25-413T=-1.


### 6.2. Example 2

In this example, a group of double integrator agents as the one in (11) is considered. The topology of the group of agents is shown in Figure 4 that was taken from [26]. The Laplacian matrix is given as(125)L=3-1-100-1-12-1000-1-13-10000-12-10000-12-1-1000-12with eigenvalues {0,1,3,4,3+2,3-2}. The experiment was carried out by considering the initial conditions x11(0)=7, x21(0)=-3, x31(0)=0, x41(0)=-2, x51(0)=2, x61(0)=3, and xi2(0)=0 for i=1,…,6. For the design of the observer λ0=0.5 and λ1=1 were considered while for the feedback law γ=1 was chosen.Figure 4
Topology for the group of double integrator agents.According to the result presented in [26], the maximum time-delay supported for the control scheme is τ∗=0.2293. To show the superiority of the solution presented in this work, a time-delay τ=0.5 s is considered that cannot be handled by the result in [26]. The prediction-based solution proposed in this work produces the position time evolution xi1 described in Figure 5 and the velocity evolution xi2 presented in Figure 6. These two figures show clearly how the consensus problem is solved. The time evolution of the control signal is shown in Figure 7. Since T1i-1=[P1i-1,0nT] and T2i-1=[0nT,P1i-1] with P1i-1=(1/6)1nT, from Theorem 12, the convergence values are x1(∞)=(7/6)1n and x2(∞)=0n.Figure 5
Double integrator case and time position evolution.Figure 6
Double integrator case and time velocity evolution.Figure 7
Double integrator case and control signals.


## 6.1. Example 1

To compare the proposed solution with a previous result, this example is inspired from [9], where three agents with the topology depicted in Figure 1 are considered. The associated Laplacian matrix takes the form(122)L=10-1-110-1-12,where λmax⁡(L)=2. Following the control strategy presented in [9], the delayed consensus problem can be solved for τ<π/2λmax⁡(L)=0.785 s.Figure 1
Three single integrator agents topology.When applying the prediction-based strategy proposed in this work, it can be shown that by considering the design parameterλ=1.0472 a maximum time-delay τ∗=1.5 s is obtained. For this experiment, τ=0.785 and the initial conditions x(0)=[-4,1,3]T are considered. The convergence to zero of the observation error ew(t-τ)=x(t)-x^(t) is shown in Figure 2(a) while the consensus of the agents state is shown in Figure 2(b). The evolution of the control signals used on the experiment is depicted in Figure 2(c).Figure 2
Single integrator case,τ=0.785 s.
(a)
Observation-prediction erroreω(t-τ)
(b)
System outputx(t)
(c)
Control signalu(t)In a more challenging situation, where the time-delay is set asτ=1.2 s, Figure 3(a) shows the time evolution of the delayed prediction error. Figure 3(b) depicts the evolution of the states and the corresponding control signals are shown in Figure 3(c). Notice how the rate of convergence for this case is slower than the previous one due to the fact that the delay is near the maximum allowable delay time.Figure 3
Single integrator case,τ=1.2 s.
(a)
Observation-prediction erroreω(t-τ)
(b)
System outputx(t)
(c)
Control signalu(t)To determine the convergence value of the agents’ state, notice that matrixP given in (73) takes the form (123)P=1-12-14112-1411234.Therefore, by considering (82), xi(t)→P1i-1x(0); that is,(124)P1i-1x(0)=0.50.250.25-413T=-1.


## 6.2. Example 2

In this example, a group of double integrator agents as the one in (11) is considered. The topology of the group of agents is shown in Figure 4 that was taken from [26]. The Laplacian matrix is given as(125)L=3-1-100-1-12-1000-1-13-10000-12-10000-12-1-1000-12with eigenvalues {0,1,3,4,3+2,3-2}. The experiment was carried out by considering the initial conditions x11(0)=7, x21(0)=-3, x31(0)=0, x41(0)=-2, x51(0)=2, x61(0)=3, and xi2(0)=0 for i=1,…,6. For the design of the observer λ0=0.5 and λ1=1 were considered while for the feedback law γ=1 was chosen.Figure 4
Topology for the group of double integrator agents.According to the result presented in [26], the maximum time-delay supported for the control scheme is τ∗=0.2293. To show the superiority of the solution presented in this work, a time-delay τ=0.5 s is considered that cannot be handled by the result in [26]. The prediction-based solution proposed in this work produces the position time evolution xi1 described in Figure 5 and the velocity evolution xi2 presented in Figure 6. These two figures show clearly how the consensus problem is solved. The time evolution of the control signal is shown in Figure 7. Since T1i-1=[P1i-1,0nT] and T2i-1=[0nT,P1i-1] with P1i-1=(1/6)1nT, from Theorem 12, the convergence values are x1(∞)=(7/6)1n and x2(∞)=0n.Figure 5
Double integrator case and time position evolution.Figure 6
Double integrator case and time velocity evolution.Figure 7
Double integrator case and control signals.


## 7. Conclusions

The delayed consensus problem is analyzed for a group of agents constituted by single and double integrators, subject to constant time-delay communication effects. To solve this problem, a protocol based on a predictor-observer scheme is proposed that allows getting estimated future values of the states of the agents that are used on the protocols that solve the consensus problems improving previously reported results. The proposed scheme has the advantage that the solution conditions depend on design parameters and are not related to the maximum eigenvalue of the Laplacian matrix associated with the considered topology as previous results. An adequate performance of the proposed solutions by means of numerical simulations that take into account time-delays values that cannot be handled by the previous solution of the problem is shown.



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*Source: 461098-2015-04-07.xml*