Abstract
This paper focuses on the consensus problem for high-order multiagent systems (MAS) with directed network and asymmetric time-varying time-delays. It is proved that the high-order multiagent system can reach consensus when the network topology contains a spanning tree and time-delay is bounded. The main contribution of this paper is that a Lyapunov-like design framework for the explicit selection of protocol parameters is provided. The Lyapunov-like design guarantees the robust consensus of the high-order multiagent system with respect to asymmetric time-delays and is independent of the exact knowledge of the topology when the communication linkages among agents are undirected and connected.
1. Introduction
In the last few years, substantial research effort from a number of researchers has been poured on the study of consensus problems for multiagent systems (MAS) due to its powerful engineering applications, such as formation control of autonomous vehicles, collective behavior of flocks, and distributed decision making in sensor networks, to name a few. The pioneering contributions in systems and control community have been made by [1, 2]. Until now, it has been proved that the consensus problem for single-integrator MAS can always be solved under certain mild conditions on the network topology [3–5].
Due to the complexity of real systems, the study on single-integrator MAS can not meet the needs of practical applications. Thus, recently, widespread interest in MAS with agents modelled by general dynamics has been excited among researchers, such as double-integrator model [6, 7] and high-order-integrator model. Specifically, high-order-integrator (or high-order) MAS have been studied in [8], where the proposed consensus protocol involves the relative information of all-order derivatives of agents’s state. Reference [9] has further extended the partial results of [8] and derived a linear-matrix-inequality-based protocol design. Reference [10] has provided a scheme to choose the coupling strength for fixed and connected topology.
In most practical networks, communication time-delays caused by limited transmission speed and distance cannot be neglected. Within the literature on consensus for MAS with time-delays, recent years have witnessed the introduction of numerous distributed protocols. According to the inducements of time-delays, these protocols can be categorized into the ones in which delays only affect the state information of the agents’ neighbors [11], the ones in which delays affect both the agents’ own state information and their neighbors’ state information [4, 7, 12–16], the ones containing distributed delays [17], and so forth. In particular, [4] has shown that the consensus of single-integrator MAS might be destabilized by large delays. By using Laplace transform technique, [7] has proved that second-order multiagent systems with bounded and constant time-delays can reach consensus. Based on nonexpansiveness of constant delay operator and Gershgorin’s circle theorem, [11] has derived a local controller for high-order MAS with diverse constant time-delays. However, time-varying communication delays are very common in MAS due to the mobility of agents and the disturbance from environment. Hence, it is necessary to study the consensus for high-order MAS with time-varying communication delays by exploring other method of protocol design.
In this paper, we investigate the consensus problem for high-order MAS with directed interactions and asymmetric time-varying communication delays. By asymmetric communication delays, we mean that the delay in communication channel from the agent to its neighbor differs from the one in the reversed channel if there are bidirectional communication linkages between them. We assume that the agent can only observe the information of the first state variable of its neighbors, and we propose a distributed protocol by applying both the instantaneous information of the agent’s own and the time-delayed relative information with respect to its neighbors. Based on a reduced-order time delay system, we derive some sufficient conditions characterized by linear matrix inequalities (LMIs), according to Lyapunov-Krasovskii functional approach. The main contribution of this paper is that we establish a Lyapunov-like design framework for the explicit selection of protocol parameters. The protocol design framework can not only guarantee the solvability of the LMIs aforementioned but also is independent of the exact knowledge of the topology when the communication linkages among agents are undirected and connected. This implies that the Lyapunov-like protocol design guarantees the robust consensus of high-order MAS with respect to asymmetric time-varying communication delays. Compared with the protocol design in [9], our design is not based on LMI but only needs to solve a simple Lyapunov equation and a simple algebraic inequality. In contrast to the literature on single-/double-integrator MAS, the results here are not simple extensions of the results therein since the protocol parameters have important effect on the consensus convergence of high-order MAS.
The remainder of this paper is organized as follows. Section 2 states the problem formulation and Section 3 presents the main results. Section 4 carries out some numerical examples and the last section provides some concluding remarks.
Notations. We let be the set of real numbers. is the -dimensional real vector space. is the set of -by- matrices with elements in . is an identity matrix. Given a matrix , denotes its spectrum (set of eigenvalues); means that is negative definite. defines a diagonal matrix with diagonal elements being . Sometimes is used to denote zero matrix. Consider . and are two index sets. denotes the Kronecker product.
2. Problem Formulation
Consider a dynamical system of autonomous agents, which are labelled through . Each agent is modelled as the following th order integrator: where is a positive integer and denotes the order of the differential equations; , and , , is the th order derivative of ; is the control input; is the initial state of agent .
The interaction/communication topology among agents can be conveniently modelled by weighted directed graph , where is the vertex set, is the arc set, and is the adjacency matrix with . An arc of , denoted by , is an ordered pair of distinct vertices of ; and are called the tail and the head of the arc, respectively. An arc if and only if . If , then we say that is a neighbor of . Denote the collection of neighbors of by . In this paper, we assume that and each element of is unique. Each vertex in represents an agent of the dynamical system (1); indicates that there is a communication linkage from agent to agent ; the element in is the weight of the linkage.
A path from to means that there is a sequence of distinct arcs in , , . A directed tree is a directed graph, where every vertex has exactly one tail except for one special vertex without any tail. We say a graph contains a spanning tree if there exists a subset of arcs such that the graph is a directed tree. A graph is said to be balanced if for each vertex the weights of its linkages satisfy , . A graph is said to be undirected if the associated adjacency matrix is symmetric. Then it is easy to see that any undirected graph is balanced. A directed graph is called strongly connected if there exists a path between any two distinct vertices of the graph; for undirected graph it is called connected. An undirected graph is called complete if for any , . The Laplacian matrix of is defined as Let with , . Then and are called the in-degree matrix of and the in-degree of vertex , respectively. From the definition, it is not hard to obtain that and . Spectral properties of the Laplacian matrix can be found in [5, 18]. Hence the details are omitted.
For the system (1), the consensus protocol is described by where , and are, respectively, the feedback gains of absolute and relative information (for convenience, we refer to the gains , , and , as the protocol parameters); piecewise continuous function is the time-varying delay affecting the communication linkage from agent to agent at time . Notice that, different from [4, 14, 19, 20], the delays in transmissions from agent to agent and from agent to agent (if there are bidirectional communication linkages between them) can be asymmetric; that is, .
Remark 1. In protocol (3), the agent is able to measure its own instantaneous state information and equipped with memories to store the signals which can be used at some future time and needs only to receive the time-delayed signals of its neighbors. The control input can be implemented by computing the instantaneous information of all-order derivatives of and the time-delayed relative information . The determination of can be carried out by assuming that the stored signals of each agent are time-stamped and neighbor transmits not only the time-delayed signal but also the time stamp. The above situation can be found in [13, 15] and also satisfied easily in practice.
Let and be the state of agent and the stacked vector of the agents’ initial states, respectively. In this paper, we are devoted to solving the following consensus problem for the MAS (1).
Definition 2. Consider the MAS (1) with some given protocol . If, for any initial state , the states of agents satisfy as for all , , then we say that the system solves a consensus problem asymptotically. In addition, if there exists such that, for any initial state , as , for all , then we say is the consensus state of the system.
In order to develop the main results, some helpful lemmas are introduced as follows.
Suppose that is a complete undirected graph of vertices and is the associated Laplacian matrix. From the definition, we have , , and the rank of is . Let be an orthogonal matrix such that , where is a diagonal matrix. Define with , . It follows that . For convenience, we let . Then and . Based on this observation and the property of Laplacian matrix, we can obtain the result below.
Lemma 3. Suppose that is a graph with the associated Laplacian matrix . Then is in the form of , where and ; . In addition, if is balanced, then ; if is strongly connected and balanced, then is positive definite.
Proof. See the Appendix: Proofs of Lemmas.
The following result can be considered as a special case of Jensen’s integral inequality given in [21].
Lemma 4 (see [21]). For any differentiable vector function and any positive definite matrix , the following inequality holds: where and .
Lemma 5 (Schur complement, see [22]). Let , , be some given matrices with appropriate dimensions and let , be symmetric; then if and only if , , or , .
3. Main Results
In this section, we first provide an equivalent condition for the consensus convergence of the systems (1) and (3) based on an orthogonal state transformation and a reduced-order time delay system. Then we give a Lyapunov-like parameter design for the protocol and prove that the maximum allowable upper bounds of time-varying delays can be determined by solving some optimization problems.
Suppose that the interaction topology of the system (1) is modelled by ; the associated Laplacian matrix is . Then the dynamics of agent can be written as where Denote by the stacked vector of the agents’ states. The closed-loop dynamics of the systems (1) and (3) are in the form of where denotes the number of different time-delays over the communication channels of the system (it is easy to get that ); for ; is defined as Next, we assume that the time-delays in (7) satisfy where . Define . The initial state of the system (7) is assumed to be , .
Remark 6. The matrix in (7) amounts to consider a Laplacian matrix that is associated with a subgraph of with all linkages affected by the same time-delay. Hence, and . According to Lemma 3, we can let , , where is given as therein. Then . Furthermore, it is important to point out that even when the topology is undirected, is unnecessarily symmetric due to . Consequently, the results of [4, 14, 19, 20] are invalid to deal with the consensus problem for MAS with asymmetric communication delays.
Applying the orthogonal linear transformation to the system (7) and denoting with and , we have where and are defined as in Remark 6. It can be seen that the system (11) is independent of the dynamics of and has the order ; the system (10) can be regarded as a forced system with the forced motion caused by the delayed state of . From and some direct computation, it follows that In addition, Proposition 7 will indicate that can be considered as the disagreement state of the system (7). Note that, for the time delay system in (10) and (11), we assume that the initial state is , , where , , and is given in (7).
We are now in a position to present an equivalent condition for the consensus convergence of the system (7).
Proposition 7. Consider the system (7) with topology . The system solves a consensus problem asymptotically if and only if each solution of the reduced-order system (11) converges to zero.
Proof. Necessity. Suppose that is any solution of the system (11) with initial state , . Let be continuously differentiable and satisfy the dynamics (10) with initial state . From (12), it follows that is a solution of the system (7) with initial state , . Denote with , . Then (12) implies that , , where is the state of agent . If the system (7) solves a consensus problem asymptotically, then as , . Hence as , , which is equivalent to . Due to and , . Hence each solution of the system (11) converges to zero.
Sufficiency. Suppose that each solution of the system (11) converges to zero, and is the solution of the system (7) with any initial state , . Let , where , . Then is a solution of the systems (10) and (11) with initial state , . The assumption implies that as . Due to (12), , in which evolves according to (10). This indicates that the system (7) solves a consensus problem asymptotically with the consensus state .
Remark 8. It should be pointed out that the orthogonal linear transformation is not uniquely defined by . Actually, any orthogonal matrix with the first column being also can derive the result of Proposition 7. In addition, the above orthogonal linear transformation can be seen as an improvement of the transformation on disagreement space which was displayed in Lemma 5.2 of [20].
Remark 9. In contrast to the transformation (7) of [15] and the tree-type transformation of [16], the order of the reduced-order system (11) induced by the orthogonal linear transformation is lower than those of the reduced-order systems induced by them (in the context that all of the three transformations are applied to the system (7)). It is easy to see that the order of the reduced-order system (11) is , whereas the order of the reduced-order system which is derived from the transformation (7) of [15] (or the tree-type transformation of [16]) is . Seen from the linear-matrix-inequality-based sufficient conditions which will be given in Theorem 12, our orthogonal linear transformation can derive lower-order linear matrix inequalities. This will reduce the computation cost to some extent when estimating the maximum allowable upper bound of time-varying delays.
Before presenting the main result, the following lemmas are introduced to give the parameter design of protocol (3).
Lemma 10. Consider and in (5). Suppose that has only one zero eigenvalue and other nonzero eigenvalues have negative real parts, and is a constant. Let with , ; is a positive definite solution of the Lyapunov equation ; is a complex number with ; . Then if and only if In particular, when , if and only if .
Proof. See the Appendix: Proofs of Lemmas.
Lemma 11. Suppose that contains a spanning tree; the associated Laplacian matrix is , ; the parameters , are chosen such that has only one zero eigenvalue and other nonzero eigenvalues have negative real parts; with , , and is a positive definite solution of the Lyapunov equation . If then is Huiwitz stable, where and is the maximum vertex in-degree over .
Proof. See the Appendix: Proofs of Lemmas.
By choosing the parameters , and as in Lemma 11 and making use of Proposition 7, we can obtain the following theorem.
Theorem 12. Consider the system (7) with fixed topology . Suppose that contains a spanning tree; the associated Laplacian matrix has the form of , where is given as in Lemma 3; the parameters , and are chosen as in Lemma 11. Then the system solves a consensus problem asymptotically if the time-varying delays satisfy , where is obtained from the following optimization problem: where and “” represents the elements below the main diagonal of a symmetric matrix.
Proof. We first show that there must exist some positive definite matrices and such that the first LMI in (15) is solvable. From the selection of protocol parameters and Lemma 11, we know that is Hurwitz stable. Hence there is a positive definite matrix such that . By Lemma 5, the first LMI in (15) is equivalent to
and is equivalent to . Consequently, if we choose , , , then for some sufficiently small , holds. This means that and satisfy the LMIs in (15).
According to Proposition 7, it suffices to prove that the zero solution of the reduced-order system (11) is asymptotically stable under the condition (15). To do this, consider the following Lyapunov-Krasovskii functional candidate:
Note that Remark 6 implies that . Then the time derivative of along the trajectory of (11) is
where is given as in (15). Let and replace with the right-hand term of (11). It follows that
where ; , , , and are given as in (15). From Lemma 5 and (15), we have . Hence there exists a positive real number such that
This proves that the zero solution of the reduced-order system (11) is asymptotically stable. Thus the conclusion holds.
Remark 13. From the proof of Theorem 12, it can be seen that the parameter design of protocol (3) in the theorem (i.e., the selection of , , and ) guarantees the solvability of the LMIs in (15). According to the nature of the parameter design (see Lemma 11), we refer to it as a Lyapunov-like parameter design.
In Lemma 11, when is undirected and connected, is positive definite. Hence , , are positive real numbers, and . In this case, for any , is Hurwitz stable. From Lemma 10, it follows that when , for any . Since is positive definite, there is an orthogonal matrix such that . Thus when , = + . In the light of the proof of Theorem 12, the inequalities in (15) hold by taking , , and sufficiently small . This result can be summarized as the following corollary.
Corollary 14. Consider the system (7) with fixed topology . Suppose that is undirected and connected; the parameters , , are chosen such that has only one zero eigenvalue and other nonzero eigenvalues have negative real parts; with , , and is a positive definite solution of the Lyapunov equation ; satisfies where is the maximum vertex in-degree over . Then the system solves a consensus problem asymptotically if the time-varying delays satisfy , where is obtained from (15) with and .
Remark 15. Corollary 14 indicates that the selection of parameters , , and is independent of the eigenvalues of Laplacian matrix but only depends on the maximum vertex in-degree of the graph. It will reduce greatly the computation and storage costs for the protocol design of practical MAS. Therefore, we can say that the parameter design in Theorem 12 is independent of the precise interaction topology when the underlying graph is connected and guarantees the robust consensus with respect to asymmetric time-varying delays for the high-order MAS (7).
Remark 16. It is worth pointing out that the results of [19] can not be applied to the high-order MAS straightforwardly, since the parameters , , and have important effect on the consensus of the system. Whereas the Lyapunov-like parameter design given in Theorem 12 can solve the consensus problem for the high-order MAS (7) very well.
Remark 17. Compared with the existing results, the main contribution of this paper is giving the Lyapunov-like parameter design which is easy to implement, independent of the precise interaction topology for the case of connected graphs, and robust with respect to asymmetric time-varying delays. Moreover, the parameter design can guarantee the existence of solution of the linear matrix inequalities given in Theorem 12, although it seems that the estimations of are conservative. More excellent estimation on the maximum allowable upper bound of time-varying delays is a commonly unsolved problem. This requires us to explore other analysis techniques which could reduce the dependence of that estimation on the knowledge of network topology.
4. Numerical Examples
Consider the system (7) of six agents with dynamics described by a triple-order integrator. The interaction topology among agents is depicted by a cycle with the arcs , , , , , . Assume that the weights of the arcs are and the communication delays affecting on the linkages are different from each other. Then the maximum vertex in-degree of the graph is , and the minimum real part of nonzero eigenvalues of the associated Laplacian matrix is .
In what follows, we design the protocol parameters , , and according to Theorem 12 (or Lemma 11).
It is not hard to obtain that the characteristic polynomial of in the system (7) is . Then any positive numbers and make satisfy the assumption in Lemma 11. Let and with , . By solving , we have , , , , and . As a result, the -by-, -by-, and -by- leading principle minors of are, respectively, , , and . Hence is positive definite for any . If we take , with , then the condition (14) becomes . Thus when and , the constraint on can be calculated as . We choose , , and for the simulation.
With the parameters chosen as above and by solving the optimization problem in (15), we can obtain that the maximum allowable upper bound of time-delays is . Assume that , , , , , and , all of which are piecewise constant functions of time . The states of agents in the system (7) asymptotically reach consensus as shown in Figure 1.
5. Conclusions
This paper has studied the consensus problem for high-order MAS with directed network and asymmetric time-varying communication delays. An equivalent condition for the consensus convergence has been established based on an orthogonal linear transformation. A Lyapunov-like protocol design has been given, which is robust with respect to asymmetric time-varying communication delays and directed interactions. The maximum allowable upper bounds of delays have been determined by solving some optimization problems. Finally, numerical examples have been worked out to demonstrate the effectiveness of the theoretical results.
Appendix
Proofs of Lemmas
Proof of Lemma 3. From the property of , let and . Then implies that Denote that and . Then is in the form of . Due to the fact that and similarity transformation does not change the eigenvalues of matrix, . It follows that . If is balanced, then . Thus . If is strongly connected and balanced, according to Theorem 7 in [4], we obtain that is a valid Laplacian matrix of an undirected graph. Then the strong connectivity of implies that the corresponding graph of is connected. Hence is positive semidefinite and its rank is . This proves that is positive definite.
Proof of Lemma 10. Denote . Since is a scalar, we can write = = = = + . The coefficient of the term in is . From the equality of the left-hand and right-hand coefficients in , we have . Notice that due to being positive definite. Then implies that .
Let . In what follows, we only need to prove that is positive definite when satisfies (13). By directly computing, we obtain Denote the -by- leading principle minor of by . For , it follows by expanding them with respect to their last rows, respectively, that
For , , we first multiply the first row and the first column by and add to the ()th row and the ()th column, respectively. Then multiply, respectively, the second, third,,th, and ()th column by , , and ; add all of them to the first column. At last, expanding the resultant determinant with respect to the first column and making use of the results of , , we have
Hence is positive definite if and only if for all . From the assumption, it is easy to see that . Since is positive definite, . As a result, . In addition, it is not hard to get if and only if . Consider . Suppose that which satisfies and . Then we have for and for . Consequently, for any , and if and only if . Next, consider . It is evident that if and only if satisfies (13). Following the similar guidelines of the proof of , we have if and only if satisfies (13). By summing up the discussion above and making use of the fact that , we obtain the first conclusion. The second conclusion follows directly from the first one.
To prove Lemma 11, we introduce the following result.
Lemma A.1 (see Theorem 4.6 in [20]). Consider and in (5). Suppose graph contains a spanning tree; the associated Laplacian matrix is , ( is given as in Lemma 3); the parameters , , are chosen such that has only one zero eigenvalue and other nonzero eigenvalues have negative real parts. If there exist some positive definite matrices , and a positive real number such that then is Hurwitz stable, where , , , is the maximum vertex in-degree over .
Proof of Lemma 11. We only need to prove that the inequalities in (A.5) hold. Take . From Lemma 10, (14) implies (A.5). The proof is completed.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant no. 61304163, the Natural Science Foundation of Shandong Province (no. ZR2013FQ008), and the Independent Innovation Foundation of Shandong University (no. 2013ZRQP006).