Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 461098 | 15 pages | https://doi.org/10.1155/2015/461098

Delayed Consensus Problem for Single and Double Integrator Systems

Academic Editor: Guangming Xie
Received18 Dec 2014
Revised16 Mar 2015
Accepted18 Mar 2015
Published07 Apr 2015

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.

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 [35]. 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 [811] 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 [1315].

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 [2325]. 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 Section 2, 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

Let be an ordered graph where(i) is a nonempty node set defined as ;(ii) is an edge set of ordered node pairs, ;(iii)an edge from is denoted as ;(iv)the grade of a node equals the number of edges that belong to the node.

A directed graph is a pair where the edge denotes that a -agent can obtain information from an -agent but not necessarily in the opposite way.

For the edge , is the parent node and is the child one. In contrast to a directed graph, a pair of nodes in an undirected graph is unordered; that is, the edge denotes that agents and can obtain information from each other. A directed path is a sequence of edges in a directed graph of the form . An undirected path in an undirected graph is defined analogously. A is a directed path that starts and ends at the same node. A directed graph is strongly 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. The adjacency matrix of a directed graph with node set is defined such that the weight is positive if and if . If the weights are not relevant, then is set equal to for all . A graph is balanced if for all . For an undirected graph, is symmetric, and thus every undirected graph is balanced.

2.2. Laplacian Matrix

The Laplacian matrix of a directed graph is given by and for all .

2.2.1. Laplacian Matrix Properties

(i).(ii)The sum of the elements of each row is zero; Consider , .(iii)For an undirected graph, is symmetric.(iv)For directed and undirected graphs, is an eigenvalue of , the sum of the elements of each row is zero, and it has an eigenvector , of dimension , corresponding to the zero eigenvalue.(v) is diagonally dominant and has no negative entries on its diagonal.(vi)All nonzero eigenvalues of are positive, while, in a directed graph, all nonzero eigenvalues of have positive real parts. Therefore, all nonzero eigenvalues of have negative real parts.(vii)Zero is always an eigenvalue of with at least multiplicity one.(viii)For an undirected graph, zero is a simple eigenvalue of if and only if the undirected path is connected.(ix)For a directed graph, zero is a simple eigenvalue of if and only if the graph has a spanning tree.(x)If the th node of is disconnected, then for all .

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 condition , it is satisfied as follows: as , for all .

3.1. Consensus for Single Integrator Agents

Consider the class of single-input single-output agents of the form for , with being the state associated with the th agent while is the corresponding input signal. Define, for a group of agents, and .

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

The closed-loop (2)-(3) produceswhere 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 tree , there exists a constant time-delay affecting the edge ; in particular, it will be considered that the time-delay is equal for all the communication channels: . Under the effects of communication time-delays, the consensus protocol (3) is modified to the delayed state causal feedback [9]: that produces the closed-loop system:where and is the state of the th agent delayed units of time.

Remark 2. It should be noticed that, under the assumption that , the delayed consensus problem can be stated by considering a set of delayed agents:with 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 formfor . 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

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] where is the Laplacian of the communication graph and is the largest eigenvalue of .

3.2. Consensus for Double Integrator Agents

As in the case of integrator agents, it will be assumed that there exist constant time-delays (corresponding to the edge ) affecting the communications channels. According to Remark 2, the effects of communication time-delays can be included on the agent dynamics in the form for with , being the state of the th agent and the corresponding input signal. is the time-delay affecting the th input signal.

The consensus for system (11) is achieved according to Definition 1; that is, and when , for all and .

System (11) can be rewritten aswhere , , and .

For the free delay case, 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], for sufficiently large positive , 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 ) for double integrator agents is presented by considering a modification of the causal feedback (13) in the formwith and as positive constants. This solution is stated for a maximum time-delay such that , withwhere is the positive root of

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 producing time-delayed closed-loop dynamics.

Suppose now that protocols (3) and (13) can be designed in terms of the future value, units of time, of the agents states, in the formfor the single integrator case and for 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, 79]. 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 and , the use of future estimated states and 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 for or equivalently producing a new state vector for the group of agents. Taking the time derivative, it is obtained that which by a forward shift of units of time produces

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 aswhere is a design constant parameter, is the estimation of , is the input signal, and is the estimation error, delayed units of time defined aswith for .

Remark 3. Contrary to a standard Luenberger observer, in the predictor-observer (23), it is not possible to use the signal since that requires future values of the state which are not available.

The convergence of the prediction error can be analyzed by considering its time evolution

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 solution of the time-delay system (26) is exponentially stable with decay rate , if there exist constants , such that

The convergence of the predictor-observer (23) is formally stated in the next result.

Lemma 5. Consider a group of 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 , signal exponentially converges to the future value for if and only if

Proof. The convergence of 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 systemsThe results presented in [36] can be used to show that system (26) will be stable for all , if and only if,In addition, to get an exponential decay rate for system (26), consider the change of variablewith , . Therefore, yieldingthat produces the decoupled set of differential-difference equationsfor .
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 such thatIn particular, by assuming that the decay rate in (31) tends to zero, implying that , it is obtained thatSince for , considering that and that , the following is obtained: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 , such thatwith decay rate , that satisfies the transcendence equation 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 variablesfor . Under these conditions,that is,

Defining the estimation errors asfor , the predictor-observer for system (11) is proposed aswhere and are positive design constant.

Lemma 7. Consider the group of double integrator agents (11) with fixed topology having a spanning tree and the predictor-observer (44). Then the estimation signals and exponentially converge, respectively, to the future values and for if and only if such thatwhere for positive design constant gains and .

Proof. Consider the time derivative of the estimation errors (43):equivalently, for . From (48) it is obtained that equivalently,The stability of system (50) can be stated by a direct analysis of its characteristic equation [36]:By considering that the crossing of the roots of (51) with the imaginary axis occurs for , it is obtained thatthat is,therefore,From (54b), which substituted in (54a) producesfrom whereEquivalently,that after some simplifications yieldsTo get a positive solution of (59), consider the auxiliary variableThen (59) is rewritten in the form with the solutionsSince and are positive constants, it is easy to see that is the only positive solution. Equations (55) and show the convergence of the estimation signals.
In order to get an exponential decay rate for system (48), consider its equivalent descriptionwhere ,and the change of variableThen, it is possible to writeand therefore obtainingNotice that if is stable, then, by (65), will be exponentially stable with decay rate . Since (63) is stable, then, by means of a continuity argument, the stability of (68) can always be established for a sufficiently small constant .

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 producing the closed-loop (9)–(69) as

Therefore, by considering the estimation error (24) system (70) can be rewritten as where, as defined previously, , , , and is the Laplacian matrix related to the topology of the agents.

By considering a nonsingular matrix defined by the generalized eigenvectors of matrix , it is possible to define the change of coordinates producing a new representation for (72a) in the formwhere is the Jordan canonical form of matrix .

The linear nature of (74) allows writing its solution as

From the properties of given in Section 2, the generalized eigenvector associated with the zero eigenvalue satisfies ; therefore, where correspond to Jordan blocks related to the independent eigenvalues , with multiplicity , , of matrix . From the structure of in (76), and [37].

To get the solution of the consensus problem, note that, in (75), and thereforeequivalently,

Equation (78) corresponds to the solution of the subsystemswith , , and . By standard properties of linear systems, notice that since , then, when , the solution of the nonautonomous system (79b) corresponds to its homogeneous solution, and therefore it is possible to assure that

It is clear now from and (78) that Since the first column of is given by , where correspond to the first row of matrix .

Additionally, from (82), it is shown that states for converge to the same valuethat depends on the agents initial conditions.

The preceding developments allow formally stating the following result.

Theorem 8. Consider a group of 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 . Moreover, .

Remark 9. In fact it is not necessary that function 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 for and is defined as

In this case, the noncausal feedback (18) can be modified as which in compact form resultswhere , , and is the Laplacian generated by the network topology.

Feedback (86), in closed-loop with system (12), produces

By considering definition (84), (43) can be expressed as , for , turning (87) in

The overall dynamic of the closed-loop (12)–(86) is obtained from (88) and (48), resulting in

Notice that, regarding (40), system (89) can be rewritten in terms of the future variables, yielding the representationwhere , for .

Taking into account from (68) that subsystem (90b) is exponentially stable, it is clear that is an equilibrium point of the subsystem (90a). Notice that is satisfied for , since the sum of the elements of each row of is always null. Notice also that the convergence of and implies the convergence of the states and due to the finite value of .

5.3. Position and Velocity Convergence

The dynamic of system (90a) can be written alternatively by considering the transformation with matrix as in (76) yielding

Since , with as the Jordan form of , then (93) can be rewritten asfor with , , and

The characteristic polynomial of the autonomous system

is obtained as

By considering the complex Jordan form with eigenvalues , , it is obtained thatfrom where it is clear that each eigenvalue of produces two eigenvalues of [7].

Lemma 10. Suppose that is an eigenvalue of the Laplacian matrix associated with a communication topology having a spanning tree of a group of double integrators agents. Then, every will produce two eigenvalues of with if and only if

Proof. From (98), the eigenvalues of can be obtained from the equationfor , in the formBy considering that with , it is obtained that Therefore, to assure , the following inequality is considered:that producesAfter some manipulations, it is obtained thatThis proves the lemma.

Remark 11. It should be noticed that Lemma 10 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 Lemma 10, the solution to the consensus protocol for double integrator agents is stated as follows.

Theorem 12. Consider a group of 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 In particular and for and defined, respectively, by the first and second row of the inverse of matrix , formed by the generalized eigenvectors of .

Proof. Taking into account the structure of the matrix given in (76), (93) can be rewritten aswhere and . From Lemma 10, subsystem (107b) is exponentially stable with an exogenous input . Therefore, since , the state convergence of system (107a) and (107b) is obtained by considering the response of the autonomous system Define now a transformation matrix formed by the generalized eigenvectors of ; . Notice that the first diagonal block of is already the Jordan block associated with the pair of zero eigenvalues of . By the uniqueness of the Jordan canonical form, up to the order of the eigenvalues, the corresponding eigenvectors are given by the first and th columns of the matrix considered in transformation (92) given as [7]The remaining eigenvectors, , will transform the second block of to the Jordan form. Therefore, by defining the transformation , the equivalent autonomous system is obtained:for with , , andIn this case, takes the formwith in a Jordan block. The time solution of (110) can be easily obtained asfrom where the evolution of the future state can be obtained in the form By considering (112), the future states converge according toNotice that since all the eigenvalues associated with are negative, the following is obtained:in addition, the general structure of allows writing Since , where and represent, respectively, the first and second row of matrix .
To complete the proof, notice that, due to relation (40), a time shift of units of time provides

The computation of the final values given in (119) can be simplified by obtaining by means of a set on independent left eigenvectors ,   of . It has been proved in [7] that and are the left eigenvectors associated with the zero eigenvalue with being a vector that satisfies and for as in (73). A direct computation of and can be done by considering which implies that

Since with , then . By the structure of matrix and condition (120), it is obtained that for . It is clear now that correspond to the first row of matrix denoted in Theorem 8 as . Direct computations show that

Therefore, the knowledge of matrices and 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 formwhere . Following the control strategy presented in [9], the delayed consensus problem can be solved for  s.

When applying the prediction-based strategy proposed in this work, it can be shown that by considering the design parameter a maximum time-delay  s is obtained. For this experiment, and the initial conditions are considered. The convergence to zero of the observation error 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).

In a more challenging situation, where the time-delay is set as  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.

To determine the convergence value of the agents’ state, notice that matrix given in (73) takes the form

Therefore, by considering (82), ; that is,

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 aswith eigenvalues . The experiment was carried out by considering the initial conditions , , , , , , and for . For the design of the observer and were considered while for the feedback law was chosen.

According to the result presented in [26], the maximum time-delay supported for the control scheme is . To show the superiority of the solution presented in this work, a time-delay  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 described in Figure 5 and the velocity evolution 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 and with , from Theorem 12, the convergence values are and .

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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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