Consensus of Switched Multiagent Systems under Relative State Constraints

The consensus problem is presented for the switched multiagent system (MAS), where the MAS is switched between continuousand 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, whichwillmake the consensus problemmuchmore 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.


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][6][7][8][9][10][11].The consensus problem was first discussed for first order [12][13][14][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][23][24][25][26][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 continuousand 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 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.

Problem Formulation
In the beginning, the graph theory will be introduced.A graph G can be defined as a three-tuple (V, E, A), where V is the node set as V = {V 1 , V 2 , . . ., V  }, E ⊆ V × V is the edge set, and A = [  ] ∈ R × , where   is the underlying weighted adjacency matrix defined as   > 0 if (, ) ∈ E and   = 0 otherwise.The set of neighbors of node V  is denoted by N  = {V  ∈ V : (V  , V  ) ∈ E}.The Laplacian matrix of the graph is defined as  = D − A, where D = [  ] ∈ R × is a diagonal matrix, with   = ∑ ∈V   .In this paper, the undirected and connected graph G will be considered, where its  is positive semidefinite and its eigenvalues are presented by 0 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   () ∈ R is the state and   () ∈ R in the control input.To achieve the consensus of the above switched MAS, the following control protocol from [32] has been proposed: where ℎ > 0 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  = 1, 2, . . ., , the following control protocol will be considered: Then, for each , the closed-loop switched MASs with state constraints are given as where the function sat  (⋅) is described as a saturated characteristic given by where  > 0 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.
Definition 1.Consider  agents, and the network of these agents is given by a graph G.The switched MAS with ( 5) and ( 6) is said to achieve the consensus problem, if for all initial conditions   (0) ∈ R,  ∈ I  .
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.

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, G() = G for any .We firstly introduce a lemma, which is used in the following sections.
According to Lemma 2, one of the main results is given as follows.
Remark 5. Note that if   = 0 (or   = 0), 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.
Proof.Similar to Theorem 3, it is proved that the average of all agents' states (1/) ∑  =1   () is invariant, which implies the final consensus state is  = (1/) ∑  =1   (0) 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  = min ∈{1,2,...,}  2 (  ) and  1 has been defined before.

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 0 < ℎ < 0.4463 can be calculated.Let ℎ = 0.3 and  = 2 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 1 and 0, where the continuous-time systems are presented by the signal 1 and the discrete-time systems are presented by the signal 0.

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

Figure 1 :
Figure 1: The structure of switching topology with 3 fixed and undirected graphs.

Figure 2 :
Figure 2: Consensus of switched systems on undirected graph with fixed topologies.

2
Complexity 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.