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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 527039, 12 pages
Consensusabilization for Continuous-Time High-Order Multiagent Systems with Time-Varying Delays
1College of Mechatronics and Automation, National University of Defense Technology, Changsha 410073, China
2Control Department, Aviation University of Air Force, Changchun 130022, China
Received 6 May 2013; Revised 4 July 2013; Accepted 25 July 2013
Academic Editor: Trung Nguyen Thoi
Copyright © 2013 Huayong Zhu 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.
For the consensus problems of high-order linear multiagent systems with time-varying delays in directed topologies, the LMI based-consensus criterion and NLMI-based consensusabilization (protocol parameters design that makes the multiagent systems achieve consensus) are investigated. Improved Lyapunov-Krasovskii functional is used for establishing the consensus convergence criteria and deriving the corresponding consensus protocol. In order to reduce the conservativeness, some proper free-weighting matrices are added into the derivative of Lyapunov-Krasovskii functional and that only keeps one necessary zoom. The numerical and simulation examples are given to demonstrate the effectiveness of the theoretical results. Compared with existing literatures, the convergence criterion and protocol design proposed have lower conservativeness.
Recently, much attention is drawn to the consensus problem in multiagent systems [1–3], due to its widely applications ranging from rendezvous , flocking , formation control  to fusion estimation . Furthermore, communication delays in consensus problems are also taken into account, for example, due to the communication congestions or retransmissions.
The study for consensus in multiagent systems with time delays started at first-order integrator systems (e.g., [8–14]), some of which were generalized to second-order (e.g., [15–23]) or high-order (e.g., ) integrator systems then. Nevertheless, the system matrix in those systems is a special companion form matrix. Therefore, it is valuable to study the consensus problems of high-order (some paper called it high dimensional) linear systems with time-varying delays.
Compared with the consensus studies on first-order, second-order, and high-order integrator multiagent systems with time delays, relatively few results were conducted in those governed by high-order linear dynamics. For linear SISO multiagent systems, the robust consensus schemes with three kinds of feedback delays were investigated in ; and the necessary and sufficient condition for the existence of consensus solution for heterogeneous high-order multiagent systems was given in . It should be noted that both of the above-mentioned studies were conducted in the frequency domain. In terms of linear MIMO multiagent systems, the time-domain methods were employed to analyze the consensus problems. In , the high-order linear MIMO consensus problems with single constant time delay under undirected communication topologies were converted into low-dimensional robust stability problems with time delay. The results showed that if the delay was less than a certain bound, the consensus protocols always existed; while  only gave the existence criteria on designing protocol parameters instead of algorithms for finding them. By projecting the state onto two subspaces (consensus subspace and consensus complement subspace), high-order linear MIMO consensus problems with single time-varying delay were equivalent simultaneous stabilization problems with low dimensions in . Therefore, a Lyapunov-Krasovskii functional was given to establish both the consensus convergence and the algorithm for protocol parameters design . However, there are still three main aspects that could be enhanced. ) Conservativeness. The criterion had relatively more conservativeness caused by the structure of the Lyapunov-Krasovskii functional and the zoom in the derivative of this functional. Constraint. All the derivatives of time delays were restricted by in . Parameters. The method of protocol design (the protocol has two parameter matrices and sometimes both of them need to be designed at once) could only calculate the second parameter matrix rather than both parameter matrices.
In this paper, we will provide free-weighting matrices  based consensus convergence criterion in high-order linear multiagent systems with single time-varying delay under directed topologies. Correspondingly, the protocol parameter matrices will be designed by solving NLMIs (nonlinear matrix inequalities). It should be noted that in undirected communicated topologies, the parameter matrices can be designed without constraining the structures for the solutions on NLMIs; while in directed topologies, some parts of those solutions should be determined in a specific form; otherwise, some nonlinear equality constraints will be added into these NLMIs, which is going to make them hard to solve. Even though we determine the structures of some solutions matrices for directed topologies, the results are better than those in , which used stronger constraints in solutions as well as a more conservative Lyapunov-Krasovskii functional.
Compared with , our results have three improvements. Conservativeness. By optimizing the structure of Lyapunov-Krasovskii functional and adding proper free-weighting matrices, our criterion has less conservativeness. Constraint. Our criterion could permit the case , even when is unknown Parameters. Both two parameter matrices can be calculated in the design of protocol could calculate directly, and for the second parameter matrix design, our method can give a feasible solution in larger time-delay bounds.
Throughout this paper, the notation represents the symmetric part in a symmetric matrix; denotes that the matrix is positive definite (positive semidefinite, negative definite, negative semidefinite); denotes the Kronecker product; can be an appropriate dimensions zero matrix or vector; for any complex vector , any real matrix and , we denote , , and , where is an identity matrix with dimensions.
2. Preliminaries and Problem Description
Let be a directed simple graph of order , where denotes the nodes set, is the edges set, and is the weighted adjacency matrix with , where if and only if edge . As is simple, holds for all . The neighbors of is defined as . The in-degree of the node are defined as . The degree matrix of is a diagonal matrix , where , . Then Laplacian matrix of is defined as .
Lemma 1 (see ). The Laplacian matrix of satisfies(1) at least has one zero eigenvalue, and is the associated right eigenvector;(2)if has a spanning tree, then 0 is a simple eigenvalue of , and all the other eigenvalues have positive real parts;(3)if does not have a spanning tree, then at least has two zero eigenvalues with the geometric multiplicity being not less than 2.
Lemma 2 (Schur Complement ). For a given symmetric matrix with the form , , , , then if and only if , or , .
In this paper, we consider a group of agents whose dynamics is described by high-order linear systems in continuous-time domain: where is the state of agent and is the consensus protocol, , .
We considered a class of protocols as follows: where , , .
Let . With protocol (2), the dynamics of multiagent systems is where is a continuously differentiable initial function.
Definition 3 (consensus). For given , the multiagent systems (3) achieve consensus if and only if for all , .
Definition 4 (consensusabilization). Multiagent systems are said to be consensusabilized by and/or if and only if the and/or make the multiagent systems achieve consensus.
3. Main Results
In this section, the consensus convergence criterion and protocol parameters design method for high-order linear multiagent systems with time-varying delays in directed topologies are provided.
Remark 6. This necessary and sufficient condition converts the consensus problem in dynamics (3) into a stability problem of dynamics (4). According to Lemma 1, if has a spanning tree, there is no . Therefore, can make it possible to stabilize systems (4); while if does not have a spanning tree, then at least has two zero eigenvalues. Without loss of generality, we assume that . Thus, (4) can be rewritten as If (4) is asymptotically stable, should be Hurwitz. In this case, multiagent systems (3) can achieve consensus even if there is no cooperation at all. Therefore, Lemma 5 is adequate to the case with or without spanning trees in communication topology. Therefore, the connectivity assumptions can be neglected.
3.1. Consensus Convergence Criteria
Theorem 7. Given the upper bound in communication delay and its derivative , multiagent systems (3) achieve consensus if there exist proper-dimensional , , , , and free-weighting matrices such that the following LMIs hold: where
Proof. Since (4) contains complex vectors and matrices, it is required to be transformed into the following form by decompositing the real and imaginary parts:
Define the following quadratic Lyapunov-Krasovs-kii functional candidates: where , , , , and , , , , , .
By Newton-Leibniz formula, for any given free-weighting matrices with proper dimensions we have where .
For any appropriate , , (13) holds
Taking the time derivative of along the trajectory of (9), it is obtained that
Adding 3 zero items into the derivative, we have
With , (18) can be zoomed; we can utilize the decomposition of in (17) as a contributor to make the zoomed (18) be quadratic forms: where . For formula (20), if and then holds for any , .
By Lemma 2, equals to . Thus, if (7), holds then (9) is asymptotically stable and multiagent systems (3) achieve consensus.
Remark 8. Theorem 7 has only one zoom in the derivative of Lyapunov-Krasovskii functional which is necessary, but Theorem 2 in  has three zooms, which make the criterion have more conservativeness. Numerical examples and Simulations, in Section 4.1, will validate that our Theorem 7 has less conservativeness.
Remark 9. Setting for all , we can get the delay-dependent and rate-independent consensus convergence criterion; while Theorem 2 in  requires that the upper bounds of are limited by .
3.2. Protocol Parameters Design Algorithm in Undirected Communication Topology
In this section, the algorithms in designing protocol parameter matrices in (2) under undirected communication topologies are investigated (including single parameter matrix design and both parameter matrices design).
Theorem 10 (double protocol parameter matrices design for undirected communication topology). Assume that the topology is an undirected graph. Given the upper bound in communication delay and its derivative , if there exist proper-dimensional , , ,
and free-weighting matrices ,
such that the following NLMIs hold:
then multiagent systems (1) can be consensusabilized by protocol (2), and the parameters are from , .
Proof. If multiagent systems (3) achieve consensus, then according to Theorem 7, inequalities (7) hold. Note that is symmetrical due to a undirected ; we can halve the dimension of and in (7). Further, can be rewritten as follows:
Then employing the congruent transformation with for in (27), we can derive where . Set , and let , , and where (note that the are half dimensional compared to those in Theorem 7). Subsequently, with and , formula (29) can be rewritten as (25). Similarly, by using congruent transformation with in dimension-halved and , respectively, inequalities (24) can be derived. Note that (24) and (25) are equivalent to (7); the protocol parameters are from , .
Remark 11. To avoid divergence of the multiagent systems, sometimes we need to design at the same time, even though it is proven in  that the is responsible for the state trajectories of those multiagent systems which achieve consensus. NLMIs (24) and (25) can be solved by using Cone Complementarity Linearization (CCL) method (see [30, 31]).
3.3. Protocol Parameters Design Algorithm in Directed Communication Topology
In this section, the algorithms in designing protocol parameter matrices in (2) under directed communication topologies are proposed.
Note that if the topology is directed, the structure of free-weighting matrices is limited by and (actually, the limitation caused by diagonal structures of ). That means even though we calculate the free-weighting matrices by solving NLMIs, we could not always acquire a diagonal structured . More importantly, both the diagonal blocks in (as well as in ) are required to be equal. This will add nonlinear equality constraints in NLMIs, which are hard to solve.
Therefore, we determine the structures of as well as in advance. Theorem 13 gives the corresponding NLMIs.
Theorem 13 (double protocol parameter matrices design for directed communication topology). Assume the topology is a directed graph. Given the upper bound in communication delay and its derivative , if there exist proper-dimensional , , , and free-weighting matrices , such that the following NLMIs hold: , where and the structures of are where , , , , , then multiagent systems (1) can be consensusablized by protocol (2), and the parameters are from , .
Proof. The process of proof is similar to that in Theorem 10. The differences are as follows: the dimension of matrices is doubled compared to those in Theorem 10; the structures of are restricted by (35) for the sake of eliminating the nonlinear equality constraints.
4. Numerical Examples and Simulations
In order to validate the theoretical results proposed in this paper, numerical examples and simulations are involved to illustrate the effectiveness. Figure 1 reveals some topologies in multiagent systems. For simplicity, assume that their adjacency matrices are limited to 0, 1 matrices.
4.1. Comparisons on Consensus Convergence Criteria
Firstly, we will compare the conservativeness of the criteria with . Additionally, the applicability for unknown in our criterion will be illustrated.
Consider multiagent systems (3) with the following parameters:
The initial function is where is the upper bound on communication delay (see formula (3)) and
Results of comparisons are shown in Table 1.
Remark 15. The comparison results illustrate that allowable upper bound on communication delays given by Theorem 7 is larger than those in  (to be more precise, ranging from 13.51% to 55.67%); the simulation results in Figure 2 validate the effectiveness. More importantly, our criterion has more advantages when becomes larger. Especially, unknown is able to judge by our criterion.
For Theorem 2 in , the number of Lyapunov-Krasovskii functional candidates is reduced to 2 with the aim of saving computing time, even though this would lead to some conservativeness. Similarly, in the current paper, we could also reduce the candidates’ number to 2 to get a higher computation efficiency. But still, for a more persuaded comparison, we select the all () Lyapunov-Krasovskii functional candidates.
4.2. Validations for Protocol Design Algorithm
Firstly, our Theorem 10 is proposed to design the parameter matrices in protocol (2) under undirected communication topologies, including calculating with a known , (the comparisons with Theorem 3 in  are made in Table 2); determining both of the parameter matrices and (while in , could not be calculated automatically by its theorems) and results are shown in Table 3.
Secondly, for directed communication topologies, the parameter matrices of protocol (2) are determined by our Theorem 13. Similarly, both cases (designing only or calculating at the same time, see Tables 2 and 3) are employed to validate the effectiveness of Theorem 13.
In this section, all the numerical examples and simulation results share the same dynamics and initial function as those in Section 4.1 (that is the multiagent systems (3) with parameters (36) and initial function (38)).
In Table 2, the parameter matrix is determined as
As we can see in this table, for undirected topology , comparisons are made between [28, Theorem 3] and our Theorem 10. Taking the situation as an example, the allowable upper bound on communication delay when designing a feasible in [28, Theorem 3] is , and the protocol parameter matrix is
However, in our Theorem 10, the allowable upper bound is , which increases one order of magnitude than that in [28, Theorem 3]. The protocol parameter matrix is and the effectiveness is validated by simulations in Figure 3.
Similarly, for directed topology , the comparisons between [28, Theorem 3] and our Theorem 13 are made in Table 2; while it should be noted that in the situation , the allowable upper bound on communication delay in our Theorem 13 is , which is less than that in [28, Theorem 3] (). However, this just holds in the case that is small. When becomes 0.9, [28, Theorem 3] cannot find a feasible any more (actually, the allowable upper bound on for [28, Theorem 3] is 0.766). Correspondingly, we can see that the allowable upper bound for [28, Theorem 3] under communication topology experiences a dramatic decrease when becomes larger. Therefore, [28, Theorem 3] is not robust enough for . By contrast, in our Theorem 13 is relatively stable when varies. For , our Theorem 13 calculates the upper bound for communication delay in the situation , while in . It seems unreasonable, but it does happen when solving the NLMIs (24) and (25). This is because the CCL algorithm is not globally optimal.
Table 3 gives the information about the results for designs under undirected communication topology (using Theorem 10) and directed communication topology (using Theorem 13), respectively. It should be noted that and are calculated at the same time rather than is determined in advance. To validate the effectiveness of our algorithms, we select the item with to carry on the simulation, and
Figure 4 shows the effectiveness in our results with time-varying delay .
We investigated the consensus problems for high-order linear multiagent systems with time varying delays, which include a consensus convergence criterion and two protocol parameters design methods (for undirected and directed communication topologies, resp.). Mainly, the following contributions were concluded in this paper:(1)a high-order linear multiagent systems consensus convergence criterion is proposed. By optimizing the structure on Lyapunov-Krasovskii functional and adding proper free-weighting matrices, the criterion has less conservativeness than the existing results;(2)the consensus convergence criterion is validated when , and even can apply with a unknown ; while  could only fit for ;(3)the protocol parameters design algorithm can give the two parameters, directly while the algorithm in  could only give but the undetermined would be set in advance;(4)under the circumstance that the is determined in advance, our theorems have less conservativeness and more robustness when the derivative of communication delay varies.
However, for the parameters designing part, the structures of matrices are constrained diagonal forms for the sake of eliminating some nonlinear inequality constraints. This has already brought some conservativeness because some feasible solutions for NLMIs are excluded manually. Therefore, the structure of will be investigated further so that our protocol parameters design could be better used in directed topologies with less conservativeness.
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