Consensus of Switched Multiagent Systems under Relative State Constraints

Wang, Qingling

doi:https://doi.org/10.1155/2017/5767495

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Research Article | Open Access

Volume 2017 | Article ID 5767495 | https://doi.org/10.1155/2017/5767495

Consensus of Switched Multiagent Systems under Relative State Constraints

Qingling Wang^1,2

Academic Editor: Michele Scarpiniti

Received31 May 2017

Accepted07 Nov 2017

Published20 Nov 2017

Abstract

The consensus problem is presented for the switched multiagent system (MAS), where the MAS is switched between continuous- and discrete-time systems with relative state constraints. With some standard assumptions, we obtain the fact that the switched MAS with relative state constraints can achieve consensus under both fixed undirected graphs and switching undirected graphs. Furthermore, based on the absolute average value of initial states, we propose sufficient conditions for consensus of the switched MAS. The challenge of this study is that relative state constraints are considered, which will make the consensus problem much more complex. One of the main contributions is that, for the switched MAS with relative state constraints, we explore the solvability of the consensus problem. Finally, we present two simulation examples to show the effectiveness of the results.

1. Introduction

Inspired by the works of [1, 2], the attention to the consensus problem of the MAS has grown in the research community. The definition of consensus implies that all the agents shall converge to a common value using a distributed interaction among these agents [3, 4]. The practical applications of consensus are diverse and can be found in many fields, such as biology, physics, control systems, and robotics [5–11]. The consensus problem was first discussed for first order [12–15] and then generalized to second order [16, 17], general linear dynamics [18, 19], and nonlinear dynamics [20, 21] in communication networks and sensor networks.

In applications, many dynamical systems interact with the environment or have physical limitations. As such, they are often affected by saturation constraints that can disrupt consensus. Therefore, the consensus problem of the MAS subject to constraints is practically important. For example, the consensus subject to input constraints was considered in [22–27], while, for the consensus with relative state constraints, only a few results were found [28, 29]. It has been known that the consensus subject to state constraints cannot be achieved due to saturations, such as the consensus with state/output saturation in [30].

It is observed that the above-mentioned MAS is focused on a single type of systems, that is, single continuous-time MAS or single discrete-time MAS. In fact, a large number of systems can be found where there exist both continuous- and discrete-time MASs simultaneously. More recently, [31] studied the stability problem of single switched systems, while the consensus of a switched MAS was presented in [32, 33]. Furthermore, there are other references concerning the switched MAS. For the switched MAS, [34] studied the finite-time control problem, while the containment consensus problem was presented in [35]. However, saturation constraints are not considered in the stability or consensus analysis of switched systems [31, 32].

Motivated by the work discussed above, we study the consensus of the switched MAS under the fixed undirected graph and the switching undirected graph. Firstly, by using some standard assumptions, the consensus shall be reached for the switched MAS under fixed undirected graph with the assumption that the absolute value of initial states is bounded. We next extend the consensus results to switched undirected graphs with a similar condition of initial states of agents. The main contributions are stated as follows: Firstly, compared with existing results [32], the challenge of this study is that state constraints are considered in the switched MAS, which increases the complexity. Secondly, the solvability of consensus problem is explored for the switched MAS with state constraints.

In what follows, we formulate the consensus problem for the switched MAS in Section 2, while the main results are illustrated in Section 3. Section 4 provides two examples to show the effectiveness, and Section 5 concludes the results of this paper.

Notation. The sign equals to when and when . Let diag be a diagonal matrix, and its entries are . implies a positive definite matrix.

2. Problem Formulation

In the beginning, the graph theory will be introduced. A graph can be defined as a three-tuple , where is the node set as , is the edge set, and , where is the underlying weighted adjacency matrix defined as if and otherwise. The set of neighbors of node is denoted by . The Laplacian matrix of the graph is defined as , where is a diagonal matrix, with . In this paper, the undirected and connected graph will be considered, where its is positive semidefinite and its eigenvalues are presented by .

In what follows, we consider the switched MAS, which are given as follows: for the continuous-time systems, and for the discrete-time systems, where is the state and in the control input. To achieve the consensus of the above switched MAS, the following control protocol from [32] has been proposed: where is the sampling period. However, in practice, the measurement part may have bounded nonlinearities or saturation constraints due to sensor limitations. Therefore, we will consider the consensus of the switched MAS with relative state constraints. That is, for each , the following control protocol will be considered: Then, for each , the closed-loop switched MASs with state constraints are given aswhere the function is described as a saturated characteristic given by where is the saturation limit. At any time instant, the activated subsystem can be chosen by a switching rule, where we consider the arbitrary switching rule in this paper. For convenience, let be .

Definition 1. Consider agents, and the network of these agents is given by a graph . The switched MAS with (5) and (6) is said to achieve the consensus problem, iffor all initial conditions .

It is worth pointing out that the switched MAS with (5) and (6) cannot achieve consensus directly. Therefore, in this paper, we will find the conditions, with which the consensus of the switched MAS with state constraints can be reached.

3. Main Results

This section is to derive some conditions for consensus of the switched MAS with fixed and switching undirected graphs.

In what follows, consensus of the switched MAS on fixed undirected graphs is considered; that is, for any . We firstly introduce a lemma, which is used in the following sections.

Lemma 2. Assume that graph is connected. Consider agent (5) or (6); the average state value is invariant.

Proof. Let . For the continuous-time subsystems (5), it is proved that . Therefore, is invariant. For the discrete-time subsystems (6), we can obtain the fact that , which implies that is invariant. This completes the proof.

According to Lemma 2, one of the main results is given as follows.

Theorem 3. Assume that graph is connected and the condition is . Then the switched MAS with (5) and (6) can achieve consensus with arbitrary switching signals ifFurthermore, the final consensus state is .

Proof. According to Lemma 2, we have the fact that is invariant, which implies the final consensus state is .
Consider a Lyapunov function for (5) and (6). It is clear that only when . That is, if and only if .
For a time period , if system (5) is switched on, we have where we have used for any , under the undirected and connected graph. Since we know , there exists some constant such that . Then, we have where .
For a period , if the discrete-time subsystem (6) is activated, the time shift of is According to the Cauchy-Schwarz inequality, we have where the fact is that the sign of is equivalent to the sign of , since . Furthermore, we define , where . Then, we have If we give a time period , where is the time for the continuous-time system (5) and is the time for the discrete-time system (6), then we have where . It follows that . Furthermore, as for , we know from (11) and (14) that or only if or . Thus, the switched MAS with (5) and (6) can achieve consensus if , can make . Note that , only when or and with . Since , then the condition that or is not satisfied. Therefore, only when is satisfied. This completes the proof.

Remark 4. With the Gershgorin disk theorem, the maximal eigenvalues of are satisfied: . Thus, the sampling period can be replaced by .

Remark 5. Note that if (or ), the consensus of the switched MAS becomes the standard consensus of single discrete-time (or continuous-time) MAS, and it is easy to show that the obtained setup (9) is correct.

In what follows, we consider the switched MAS with (5) and (6) on switched and undirected graphs. That is, the graph will be randomly switched among distinct and finite topologies , and if and only if the random switching signal . Then, the following results can be obtained.

Theorem 6. Assume graph is undirected and connected for each and the condition is , where the Laplacian matrix of graph is . Then, the switched MAS with (5) and (6) can solve the consensus problem if Furthermore, the final consensus state is .

Proof. Similar to Theorem 3, it is proved that the average of all agents’ states is invariant, which implies the final consensus state is if the consensus is reached.
In view of the proof of Theorem 3, for a time period , if system (5) is switched on, we have where and has been defined before.
For a time period , if the discrete-time system (6) is switched on, the time shift of is shown as since . It is clear that if and only if . Furthermore, we have where has been defined in Theorem 3. For any , we have where . It follows that . Then, similar to the proof of Theorem 3, since , we have or only when . Thus, the switched MAS with (5) and (6) can achieve consensus, which completes the proof.

4. Simulations

This section will present two examples to illustrate the results on the fixed and switching undirected graphs.

Example 1. Let the fixed undirected graph be given as Figure 1(a). According to Theorem 3, the fact that sampling period condition is can be calculated. Let and in this example. Figure 2 shows the law of the switched MAS and the trajectories of agents under different initial conditions. In Figure 2, the switching signals of switched systems are presented by and , where the continuous-time systems are presented by the signal and the discrete-time systems are presented by the signal . Figure 2(a) is with the initial states , and Figure 2(b) is with the initial states . Under the initial states , the absolute final value is , which satisfies condition (9), and then the switched MAS can achieve consensus under arbitrarily given switching law. However, under the initial states , the absolute final value is in Figure 2(b), which does not satisfy condition (9), and thus the switched MAS cannot achieve consensus.

(a)

(b)

(c)

(a)

(b)

Example 2. Suppose that the switching undirected graph is given in Figure 1, and the switching set . The sampling period condition can be calculated as . Let and . In the top panel of Figure 3, we give the law of switching topologies, where the signals 1, 2, and represent the graphs , and , respectively. The law of switched systems is shown in the middle panel of Figure 3. In the bottom panel of Figure 3, we give the trajectories of agents with the initial states . The figure shows that, with the condition in Theorem 6, the switched MAS can achieve consensus.

5. Conclusions

In this study, the consensus was studied for the switched MAS with relative state constraints. Since saturation constraints have been considered, the switched MAS should satisfy some conditions for achieving consensus. Therefore, considering the switched MAS with saturations, we have proposed the conditions for fixed and switching undirected graphs. Furthermore, the general magnitude saturation function has been considered. Then, by employing some standard assumptions, sufficient conditions have been provided for the switched MAS with state constraints. Finally, we have presented two examples to verify the main results.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (61503079), the Jiangsu Natural Science Foundation (BK20150625), the Fundamental Research Funds for the Central Universities, and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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Copyright

Copyright © 2017 Qingling Wang. 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.

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