Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 192351, 13 pages

http://dx.doi.org/10.1155/2013/192351

Research Article

State Consensus Analysis and Design for High-Order Discrete-Time Linear Multiagent Systems

Yanrong Ge,1,2 Yangzhou Chen,¹ Yaxiao Zhang,¹ and Zhonghe He¹

¹College of Electronic Information and Control Engineering, Beijing University of Technology, Beijing 100124, China

²College of Physics Science and Information Engineering, Hebei Normal University, Shijiazhuang 050024, China

Received 17 April 2013; Accepted 17 June 2013

Academic Editor: Guanghui Wen

Copyright © 2013 Yanrong Ge et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The paper deals with the state consensus problem of high-order discrete-time linear multiagent systems (DLMASs) with fixed information topologies. We consider three aspects of the consensus analysis and design problem: (1) the convergence criteria of global state consensus, (2) the calculation of the state consensus function, and (3) the determination of the weighted matrix and the feedback gain matrix in the consensus protocol. We solve the consensus problem by proposing a linear transformation to translate it into a partial stability problem. Based on the approach, we obtain necessary and sufficient criteria in terms of Schur stability of matrices and present an analytical expression of the state consensus function. We also propose a design process to determine the feedback gain matrix in the consensus protocol. Finally, we extend the state consensus to the formation control. The results are explained by several numerical examples.

1. Introduction

In recent years, the consensus problem of multiagent systems (MASs) has been becoming a significant research topic because of its broad practical applications, including the work load balance in a network of parallel computers [1], the clock synchronization [2], distributed decision [3], consensus filtering and estimation in sensor networks [4–6], rendezvous, and the formation of various moving objects [7–11] such as underwater vehicles, aircrafts, satellites, mobile robots, and intelligent vehicles in automated highway systems, to name only a few. Hence, its study has captured attention of the researchers from different disciplines.

MASs are comprised of locally interacting agents equipped with dedicated sensing, computing, and communication devices. The consensus problem of MASs is to design a distributed control law for each agent, using only information from itself and its neighbors, such that all agents achieve an agreement on some quantities of interest. To design and analyse this class of systems, one needs to consider three essential elements: a dynamic model describing the states of the agents, which can be either continuous time or discrete time, linear or nonlinear, homogeneous or heterogeneous, time varying or time invariant, low order or high order; an information topology describing communication network between the agents, which can be either undirected or directed, fixed or switched; a protocol (control input) for each of the agents describing how the agents interact on each other according to the given information topology, which can be synchronous or asynchronous, with or without time delay.

Up to now, numerous researches have been done for continuous-time MASs in different settings from the above cases [10, 12–18]. This paper focuses on the study of high-order discrete-time linear multiagent systems (DLMASs) by proposing a linear transformation to translate the consensus problem into a partial stability problem. Although this approach can be extended to any setting from the above cases, we pay our attention only to the case of fixed information topology and in the absence of time delay for giving prominence to the trait of the approach. Here we give an overview mainly to the DLMASs.

Reference [19] first proposed an interesting model for self-propelled particle systems, where all agents move in a plane with the same speed but different headings, and showed that in the model all agents might eventually move in the same direction despite the absence of centralized coordination. Reference [20] further gave a mathematically rigorous qualitative analysis. Then, a theoretical explanation was given for the consensus behavior of Vicsek model on the basis of graph theory in [21, 22]. A necessary and sufficient condition was given for the average consensus criterion in [23]. Reference [24] further considered the case of switching network topologies for the average consensus. The average consensus was investigated for the systems with uncertain communication environments and time-varying topologies in [25] and with communication constraints in [26]. Reference [27] presented convergence results for the time-varying protocol in the absence or presence of communication delays. Reference [28] proposed an asynchronous time-varying consensus protocol. Reference [29] further discussed nonlinear systems with time-dependent communication links. References [30, 31] addressed the case with both time-varying delays and switching information topologies and provided a class of effective consensus protocols by repeatedly using the same state information at two time steps.

The researches mentioned above were limited to first-order systems. The extension to second-order systems was done for the systems with time-varying delays and time-varying interaction topology in [32], and for the systems with nonuniform time-delays and dynamically changing topologies in [33].

Recent researches were turned to the high-order DLMASs in [34–39]. Reference [34] studied a class of dynamic average consensus algorithms that allow a group of agents to track the average of their reference inputs. Reference [35] proposed an observer-type protocol based on the relative outputs of neighboring agents. Reference [36] studied the convergence speed for the high-order systems with random networks and arbitrary weights. Reference [37] addressed the high-order systems with or without delays. These researches were focused on the consensus convergence criteria for the proposed protocols. Another significant topic is the design of the gain matrices of the protocols in [38, 39].

This paper deals with both analysis and design problems of the state consensus for general high-order DLMASs. Compared with the existing works, the contributions of the paper are summarized as follows. Firstly, motivated by [12], we improve the protocol by adding a self-feedback of the agent to achieve the expected consensus dynamics, whereas [13] introduced the internal model to change the given dynamic to achieve the expected consensus dynamics and [14, 15] introduced the virtual leader to guide the multiagent systems to achieve the expected consensus dynamics. Secondly, we propose a state linear transformation to translate the consensus problem into a partial stability problem. The approach is motivated by the error variable method or the state space decomposition method in [12, 16]. However, our improvement can more spontaneously and conveniently deal with various settings of the consensus problems. Based on the partial stability theory, we educe new necessary and sufficient consensus convergence criteria in terms of stability of matrices and moreover give an explicit analytical expression of the state consensus function based on the different contributions of the initial states of the agents and the protocols. Thirdly, based on stability theorem, we give a design procedure to determine the gain matrices in the protocol on the basis of algebraic Riccati inequality similarly to [38, 39]. Fourthly, we extend the state consensus results to the formation control problem.

The remainder of the paper is organized as follows. Section 2 introduces some basic concepts and notations, and formulates the problem under investigation. Section 3 firstly introduces a linear transformation which translates the consensus problem of the multiagent systems into a partial stability problem of the corresponding transformed system, and then educes a new necessary and sufficient condition for the multiagent system to achieve global state consensus and presents an analytical expression of the state consensus function. Section 4 shows a design procedure to determine the gain matrices in the state consensus protocol. Section 5 extends the approach for the analysis and design of the state consensus to the formation control problem. Section 6 gives numerical examples to explain the theoretical results. Section 7 concludes the paper. All the proofs of the results are deposited in the appendix for the sake of reading.

2. Problem Description

Before stating the consensus problem, we give some basic concepts and notations. Let and be the sets of real matrices and complex matrices, respectively. Matrices, if not explicitly stated, have appropriate dimensions in all settings. The superscript “” means transpose for real matrices, and the superscript “” means conjugate transpose for complex matrices. presents the identity matrix of dimension , and sometimes is used for simplicity. denotes the vector of dimension with all entries equal to one. is applied to denote zero matrices/vectors of any size, with zero components. A matrix is said to be Schur stable if all of its eigenvalues have magnitude less than . The Kronecker product is denoted by and the Hadamard product by in [40]. The following properties of the Kronecker product will be used: ; ; .

We consider DLMASs with homogeneous agents and assume they are described by

where , , is assumed to be stabilizable, is the state of the current time , denotes the state at the next time , and is the control input of the current time .

The control input will be constructed based on the available information of the agent . Let denote the index set of the agents which can send their state information to the agent . We call the set the information topology of the DLMASs (1). It is well known that one can use a digraph to express the information topology , where is the index set of agents, is the set of directed edges to describe the information interaction between agents; that is, . Based on the directed edges, one can construct an adjacency matrix , whose entries are defined as for , for , and for . The corresponding in-degree matrix and graph Laplacian are defined as and , respectively, where is the in-degree of the vertex . A directed spanning tree of the digraph is a tree covering all the vertices of the digraph. The following results are well known.

Lemma 1 (see [22]). The Laplacian matrix has the following properties: (1) all of the eigenvalues of are either in the open right half complex plan or equal to ; (2) is a simple eigenvalue of if and only if the digraph contains a directed spanning tree.

Given the information topology , we construct the following linear consensus protocol:

where , are feedback gain matrices to be determined, which are relative to the consensus state and the convergence rate, respectively. is a weighted matrix associated with the information topology . For the sake of expression, we also define the weighted adjacency matrix by using Hadamard product of matrices and the weighted Laplacian with weights , where is the corresponding weighted in-degree matrix with weighted in-degrees .

Definition 2. For the given information topology , the DLMASs (1) are said to achieve global state consensus via the protocol (2) if for any given initial state , , there exists an -dimensional vector function depending on the initial states such that . The function is called a state consensus function.

In this paper, we will address the following three aspects of the state consensus problem: (i) to give criteria of global state consensus, that is, for any given information topology , weighted matrix and feedback gain matrices and to find the conditions of the DLMASs (1) achieving global state consensus via the protocol (2); (ii) to calculate the state consensus function if the DLMASs (1) achieve global state consensus via the protocol (2); (iii) to determine the matrices and such that the DLMASs (1) achieve global state consensus via the protocol (2).

First of all, we transform the state consensus problem to the partial stability problem. Then, based on the partial stability theorem framework, we educe new necessary and sufficient consensus convergence criteria and state a procedure to determine the gain matrices in the protocol on the basis of algebraic Riccati inequality. We also give an explicit analytical expression of the state consensus function based on the respective contributions of the initial states and the protocols. Finally, we extend the results to formation control.

3. State Consensus Analysis

In this section, we first introduce a linear transformation which translates the consensus problem of the multiagent systems into a partial stability problem of the corresponding transformed system. Then, we educe a necessary and sufficient condition for the DLMASs (1) to achieve global state consensus via the protocol (2), and present an analysis expression of the state consensus function. Finally, we discuss some interesting remarks and corollaries based on the result.

Let . The dynamics of the DLMASs (1) with the protocol (2) is described by

where

We propose a state linear transformation for the linear system (3) as follows:

where the block matrix is defined as

And the matrix ,  , is chosen such that the following two conditions are satisfied:

(1)the row vectors in each of the matrices are linearly independent, respectively;(2)the identities , , are held.

Lemma 3. The inverse of the matrix admits the following form:

where , , , are the blocks indefinitely described.

Using the linear transformation (5), we transform the linear system (3) into the following system:

or the form of two equations

where , , and .

We show that the state consensus problem of the DLMASs (1) with the protocol (2) can be transformed into a partial stability problem.

Definition 4 (see [41]). The linear system (8) is said to be asymptotically stable with respect to (or asymptotically -stable in short) if for any bounded initial state of the system (8).

Lemma 5. Under the given information topology , the DLMASs (1) achieve global state consensus via the protocol (2) if and only if the equilibrium point of the linear system (8) is asymptotically -stable. Moreover, the state consensus function of the agents is .

Lemma 5 builds a bridge between the consensus problem and the partial stability problem. Now we focus on the asymptotical -stability of the linear system (8). We can verify the following lemma.

Lemma 6. The system (9) is of the following form:

where , , and .

Combining Lemma 5 with Lemma 6, we directly get the following theorem.

Theorem 7. Under the given information topology , the DLMASs (1) achieve global state consensus via the protocol (2) if and only if matrix in (10) is Schur stable. Moreover, the state consensus function is

Subsequently, we give some interesting remarks and corollaries based on the result.

Remark 8. Since , the result of Theorem 7 is in fact independent of the choice of the matrix although both and formula (11) in Theorem 7 contain and . Hence, for simplicity, we take it in the following form:

The corresponding inverse matrix is

Thus, we can write and into

Corollary 9. Under the given information topology , the DLMASs (1) achieve global consensus via the protocol (2) if and only if all the matrices are Schur stable, where , , are the eigenvalues of the matrix . Moreover, the state consensus function is expressed by

where satisfies and .

One can verify that as the state consensus functions in formulas (11) and (15) are the same.

Remark 10. From Schur stability of in the formula (14), we can conclude that if is not Schur stable, it is a necessary condition of the consensus that the digraph expressing the information topology has a directed spanning tree. In fact, since the condition of directed spanning tree is equivalent to Hurwitz stability of , a lack of directed spanning tree means that has a zero eigenvalue. In this case, we transform into its Jordan form via the matrix , where is the matrix such that is the Jordan form, and thus we have

One can verify that the eigenvalues of are the members of the eigenvalues of , and thus is not Schur stable if is not Schur stable. On the other hand, if is Schur stable, one can take to make the DLMASs (1) achieve global consensus, which implies that for any initial states all the agents always converge to the equilibrium point .

Hence, from the formula (16) we can educe another global consensus criterion.

Corollary 11. If is not Schur stable, then under the given information topology , the DLMASs (1) achieve global state consensus via the protocol (2) if and only if has eigenvalues with positive real part , , and the matrices , , are Schur stable.

Remark 12. If the state consensus function in (11) is a constant vector equal to the average of the initial states of all the agents, the consensus is called the average consensus. From the formula (11) we educe the following result on the average consensus.

Corollary 13. The DLMASs (1) achieve global average consensus via the protocol (2) if and only the matrix is Schur stable and .

If , then the last condition in Corollary 13 becomes , or equivalently, the digraph is either undirected connected or directed strong connected and balanced. More specially, if is a symmetric matrix (equivalently, the digraph becomes undirected connected), the condition is satisfied and thus the average consensus is achieved.

Remark 14. When , , and , the DLMASs (1) are called a single-integrator one. In this case, and . We educe the following result.

Corollary 15. Under the given information topology , the single-integrator DLMASs (1) achieve global state consensus via the protocol (2) if and only if the following two conditions are held simultaneously: (1) the matrix is Hurwitz stable; that is, the digraph admits a directed spanning tree; (2) the products , , , are in the open unit circle of the complex plane with the centre at , where , , are the eigenvalues of the matrix and , , are the eigenvalues of the matrix . The corresponding consensus function (11) becomes a constant vector

Moreover, the single-integrator DLMASs (1) achieve global average consensus via the protocol (2) if and only both of the above conditions are satisfied and in addition .

Remark 16. When , , and , the DLMASs (1) are called a double-integrator one, whose state vector can be seen as consisting of the position and velocity in the dimensional space .

Corollary 17. Under the given information topology , the double-integrator DLMASs (1) achieve global state consensus via the protocol (2) if and only if is Schur stable. Moreover, the consensus function is

The consensus function above can be decomposed into the position consensus function

which is a linear function of discrete time , and the constant velocity consensus function is as follows:

Similarly, we can define the velocity average consensus, that is, if the DLMASs (1) achieve global consensus via the protocol (2), and the velocity consensus function is a constant vector equal to the average of the initial velocities of all the agents.

Corollary 18. Under the given information topology , the double-integrator DLMASs (1) achieve global velocity average consensus via the protocol (2) if and only the matrix is Schur stable and .

It is obvious that if , then ; that is, the last condition in Corollary 18 is satisfied, and thus, the state consensus function becomes

4. Design of Gain Matrices

In this section, we discuss the third problem, that is, how to determine the weighted matrix and the gain matrices and , such that the DLMASs (1) achieve global state consensus via the protocol (2).

Theorem 7 shows that the matrices , , and should be taken to ensure that the matrix is Schur stable. Furthermore, from Corollary 11 we see that if the matrix with respect to the information topology has been given, we need only to design the gain matrices and to ensure that the matrices are Schur stable, where , , are the eigenvalues of the matrix . The matrix is often taken to obtain an expected consensus dynamics. The matrix is designed to achieve state consensus and expected convergence rate. Its design needs the following lemma.

Lemma 19 (see [38]). Supposing that the matrix is not Schur stable but is stabilizable, then there is a critical value , such that for any number with the modified Riccati inequality

admits a positive definite matrix solution , where depends on the unstable eigenvalues of the matrix .

We define functions and . Motivated by [38], we get the following theorem.

Theorem 20. Supposing that the matrix is stabilizable, the gain matrix has been taken such that the expected consensus dynamic matrix is not Schur stable, and the weighted matrix with respect to the information topology is given such that is Hurwitz stable with eigenvalues , ; then, for the DLMASs (1) to achieve state consensus via the protocol (2), the matrix can be designed as , where is an arbitrary constant satisfying , is a critical value which depends on the unstable eigenvalues of the matrix , and is a solution of the algebraic Riccati inequality (22).

Based on Theorem 20, we give the following algorithm of determining the feedback gain matrices and in the protocol (2).

Algorithm 21. Design procedure of the gain matrices and .

Step 1. Verify the stabilizability condition of and the spanning tree condition of the information topology . If neither of them is satisfied, then stop. Otherwise, design the weighted Laplacian such that is Hurwitz stable with eigenvalues , .

Step 2. Design such that is the matrix of the expected consensus dynamics of the DLMASs (1) and is not Schur stable.

Step 3. Calculate all the eigenvalues of , which are composed of the stable eigenvalues , and unstable ones , , .

Step 4. Calculate the critical value . If is invertible, then . If is of rank one, then . Otherwise, apply Wonham decomposition to the unstable part of to convert the multiple input system to single input subsystems, where is the number of the Jordan blocks of matrix . Specifically, there is a nonsingular real matrix with a compatible dimension such that and take the form

where the symbol denotes possibly nonzero parts and with and for all is controllable and . In this case, is lower bounded by , where the index is defined by and is the Jordan block of the unstable part of matrix .

Step 5. Calculate the value such that .

Step 6. Solve (22) with for a positive definite matrix .

Step 7. Calculate the matrix .

5. Application to Formation Control

In this section, the consensus approach is modified to solve the formation control problem of the DLMASs (1). Let describe a constant formation of the agent network in a reference coordinate frame, where is the formation variable corresponding to the agent . The variable denotes the relative formation vector between the agents and , which is assumed to be independent of the reference coordinate.

We modify the consensus protocol (2) and propose a distributed formation protocol as follows:

Definition 22. Under the given information topology , the DLMASs (1) achieve the given formation via the protocol (24) if as , for all , that is, if there is a function such that as , for all , where is called reference state consensus function.

Theorem 23. Under the given information topology , the DLMASs (1) achieve the formation via the protocol (24) if and only if the matrix is Schur stable and . Moreover, the reference state consensus function is

Similarly to Corollary 11, we get the following corollary for the formation control.

Corollary 24. Under the given information topology , the DLMASs (1) achieve the formation via the protocol (24) if and only if all the matrices are Schur stable and , where , , are the eigenvalues of the matrix .

Remark 25. Note that not all kinds of formation structure can be achieved for the DLMASs (1) by using the protocol (24). The achievable formation structures have to satisfy the constraints . The formation protocol (24) for a given achievable formation can be constructed analogously by using the Algorithm 21 in Section 4.

6. Numerical Examples

In this section, we give some illustrative examples.

Example 26 (state consensus analysis). We consider the DLMASs (1) consisting of four agents described by the following matrices:

Supposed that we are given the information topology , the weights , and the gain matrices and , we are required to verify the consensus convergence.

The matrix is Schur stable since its eigenvalues are , ,,, and . Therefore, according to Theorem 7, the DLMASs (1) with the matrices in (26) achieve global consensus via the protocol (2) under the given information topology and the gain matrix , . For the initial states , , , and , the corresponding state consensus function in (11) is . Figure 1 shows the state trajectories of the agents and the trajectory of the state consensus function marked by circles.

Figure 1: State trajectories of DLMASs.

Example 27 (gain matrices design for a formation control). The consensus problem of multiagent systems has many practical applications, such as formation control of mobile robots and cooperative control of unmanned airborne vehicles. In this example, we consider that DLMASs consisting of four mobile robots are described as

where , , and are the position, the velocity, and the acceleration input of the robot , respectively. The information topology and the weights are the same as in Example 26. The eigenvalues of the matrix are , , and . We choose , , , and for the formation of the mobile robots. Note that in (27) is stabilizable. The eigenvalues of matrix are (4 multiples). We assume that the expected consensus dynamics of the agents is not changed, and thus, we have . It is easy to verify that the formation satisfies the constraints . One can get through Step 4 of the Algorithm 21. By Step 5, one can obtain , and the corresponding . The modified Riccati inequality (22) admits a solution

The corresponding control gain is calculated as follows:

The eigenvalues of matrix are (2 multiples), (2 multiples), (2 multiples), and (2 multiples), that is, the matrix is Schur stable. By Theorem 23, the system (27) via the protocol (24) achieves the formation .

For the initial states , , , and , we get the reference state consensus function according to (25). Figure 2 shows the position and velocity trajectories of the mobile robots and the trajectory of the reference state consensus function marked by dots, and Figure 3 shows the formation evolution trajectories. From Figure 3 it is clear to see that the mobile robots achieve the expected formation, where the symbol diamond denotes the initial position, and the symbol circle denotes the position at the moment of the mobile robots, respectively.

Figure 2: Position and velocity trajectories of mobile robots.

Figure 3: Formation of four mobile robots.

7. Conclusions

We considered the state consensus problem of high-order discrete-time linear multiagent systems with fixed directed information topology. A linear transformation approach was proposed to translate the consensus problem of multiagent systems into a partial stability problem of the corresponding transformed systems. We have shown that the approach is powerful in dealing with the three aspects of the consensus problem: the criteria of global state consensus, the calculation of the state consensus function, and the determination the weighted matrix and the feedback gain matrix. Precisely, we have educed new necessary and sufficient consensus criteria in terms of Schur stability of a matrix related to the weighted Laplacian matrix and presented an analytical expression of the state consensus function. In addition, we have stated a design process of determining the feedback gain matrix under the condition of each agent being stabilizable. The consensus algorithm has been further applied to solve the formation control problem of multiagent systems.

Though the work in this paper focuses on the high-order discrete-time linear multiagent systems with fixed information topology and without time delay, it is undoubted that the approach can be easily extended to more complex cases, which will be dealt with in the future works.

Appendix

Proof of Lemma 3. Assuming the inverse to be with columns ,  , we prove that the equality is correct. Since the matrix is invertible and thus each column of can be linearly represented by the columns of the matrix , so the matrix can be represented by , where each column of the matrix can be linearly represented by matrices , , and by those of the matrix . Left multiplying by gets . Because of , one has , , which implies . On the other hand, from the equalities , , one gets , , which means . Hence, from the equality , one deduces , that is, and thus . In other words, one can write into , where is a matrix of the order . From the identity , one has , and thus, from one can get , that is, . Finally, one has the expression .

Proof of Lemma 5. In fact, if there is such that , , it follows that , , in virtue of , , and therefore the necessary has been proved. Conversely, by virtue of Lemma 3, one can verify , . So from , , it follows that , , where , and thus, the sufficiency has been verified.

Proof of Lemma 6. By observation, we only need to show that , , and . In fact, since , , and , we have that

Thus, system (9) becomes

Denoting , , and , where , we get (10).

Proof of Theorem 7. The necessary and sufficient condition is verified directly by using the Lemmas 5 and 6. Now we focus on the calculation of the state consensus function. The first equation in (10) gives . From the second equation in (10) we have

Substituting , , and into the above formula, one gets . Thus, by Lemma 5 one has the state consensus function in (11).

Proof of Corollary 9. First of all, one easily verifies that the Schur stability of is equivalent to Schur stability of all the matrices by transforming into its Jordan form. We focus on the calculation of the state consensus function. Rewrite the system (3) as . So for the left eigenvector with the property of the Laplacian with respect to the zero eigenvalue, we obtain . When the consensus is achieved, we have that and thus the state consensus function is , which can be written into the form (15).

Proof of Corollary 15. In this case, the two conditions in Corollary 11 become: the matrix is Hurwitz stable, that is, the digraph admits a directed spanning tree; for all the eigenvalues , , of the matrix , all the matrix are Schur stable. Let , be the eigenvalues of the matrix . Hence, the Schur stability of the matrix is equivalent to that the products , , , are in the open unit circle of the complex plane with the centre at .

Now we calculate the state consensus function. In this case, the state consensus function becomes . Since is Schur stable, instead of the previous state consensus function , we take the following state consensus value . Since , , and , and noticing that , the previous state consensus value can be written into (17).

Proof of Corollary 17. Since

Equation (11) becomes

Letting

and , we get

Thus, we have the following:

Substituting it into (A.5), we get

If is Schur stable, the consensus function above can be replaced by

Proof of Theorem 20. Supposed that the matrices and have been given, by Corollary 11, we need only to verify that the matrix ensures that all the matrices are Schur stable, where , , are the eigenvalues of the matrix .

It is clear that there exists such that , . We can verify that the following systems , , admit a common Lyapunov function . In fact, let , then we have that

That is, all the matrices , , are Schur stable.

Proof of Theorem 23. Let , . Then the DLMASs (1) reach the formation if and only if as , for all . So the formation problem on the variables is transformed to the consensus problem on the variables .

From (1), one gets , . Substituting the protocol (24) and writing it into the vector form, we have that

Introducing the state linear transformation for the linear system (A.12), one obtains

Let , then

is held. Since must be the equilibrium point if the DLMASs (1) reach the formation , one has . The residuary proof is similar to that of Theorem 7.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (61079001 and 61273006), National High Technology Research and Development Program of China (863 Program) (2011AA110301), Specialized Research Fund for the Doctoral Program of Higher Education of China (20111103110017), Hebei Province Science and Technology Research and Development Planning Project (10203548D), and Hebei Province Science and Technology Conditions Building Program (11963546D).

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