# Consensus of Switched Multiagent Systems under Relative State Constraints

Authors: Qingling Wang
Journal: Complexity
 (2017)

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## 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.


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## Body

## 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 Section2, 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 signy equals to +1 when y≥0 and -1 when y<0. Let diag{ω1,ω2,…,ωn} be a diagonal matrix, and its entries are ωi,i=1,2,…,n. P>0 implies a positive definite matrix.


## 2. Problem Formulation

In the beginning, the graph theory will be introduced. A graphG can be defined as a three-tuple V,E,A, where V is the node set as V=v1,v2,…,vN, E⊆V×V is the edge set, and A=aij∈RN×N, where aij is the underlying weighted adjacency matrix defined as aij>0 if i,j∈E and aij=0 otherwise. The set of neighbors of node vi is denoted by Ni=vi∈V:vi,vj∈E. The Laplacian matrix of the graph is defined as L=D-A, where D=dij∈RN×N is a diagonal matrix, with dii=∑j∈Vaij. In this paper, the undirected and connected graph G will be considered, where its L is positive semidefinite and its eigenvalues are presented by 0=λ1<λ2≤⋯≤λN.In what follows, we consider the switched MAS, which are given as follows:(1)x˙it=uit,i=1,2,…,N,t∈R+, for the continuous-time systems, and (2)xit+1=xit+uit,i=1,2,…,N,t∈N, for the discrete-time systems, where xi(t)∈R is the state and ui(t)∈R in the control input. To achieve the consensus of the above switched MAS, the following control protocol from [32] has been proposed: (3)uit=∑j∈Vaijtxjt-xit,i=1,2,…,N,t∈R+h∑j∈Vaijtxjt-xit,i=1,2,…,N,t∈N, 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 i=1,2,…,N, the following control protocol will be considered: (4)uit=∑j∈VaijtsatMxjt-satMxit,t∈R+h∑j∈VaijtsatMxjt-satMxit,t∈N. Then, for each i, the closed-loop switched MASs with state constraints are given as(5)x˙it=∑j∈VaijsatMxjt-satMxit,t∈R+,(6)xit+1=xit+h∑j∈VaijsatMxjt-satMxit,t∈N,where the function satM· is described as a saturated characteristic given by (7)satMx=x,ifx≤MMsignx,ifx>M, where M>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. For convenience, let i=1,2,…,N be i∈IN.Definition 1.
ConsiderN 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(8)limt→∞xit-xjt=0,for all initial conditions xi(0)∈R,i∈IN.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,G(t)=G for any t. We firstly introduce a lemma, which is used in the following sections.Lemma 2.
Assume that graphG is connected. Consider agent (5) or (6); the average state value 1/N∑i=1Nxi(t) is invariant.Proof.
Letξ(t)=1/N∑i=1Nxi(t). For the continuous-time subsystems (5), it is proved that ξ˙(t)=1/N1NTLsatM(x)=0. Therefore, ξ(t) is invariant. For the discrete-time subsystems (6), we can obtain the fact that ξ(t+1)=1/N∑i=1Nxi(t)-1/Nh1NTLsatM(x(t))=ξ(t), which implies that ξ(t) is invariant. This completes the proof.According to Lemma2, one of the main results is given as follows.Theorem 3.
Assume that graphG is connected and the condition is 0<h<1/maxi∈IN∑j∈Vaij. Then the switched MAS with (5) and (6) can achieve consensus with arbitrary switching signals if(9)1N∑i=1Nxi0≤M.Furthermore, the final consensus state is limt→∞xi(t)=1/N∑i=1Nxi0.Proof.
According to Lemma2, we have the fact that ξ(t)=1/N∑i=1Nxi(t) is invariant, which implies the final consensus state is ξ=1/N∑i=1Nxi(0).
Consider a Lyapunov function(10)Vt=∑i=1Nxi2t≥0, for (5) and (6). It is clear that V(t)=0 only when xi(t)=0,i∈IN. That is, V(t)=0 if and only if xi(t)=xj(t).
For a time periodtc, if system (5) is switched on, we have(11)V˙t=2∑i=1Nxitx˙it=2∑i=1Nxit∑j∈VaijxitsatMxjt-satMxit=-∑i=1N∑j∈Vaijxit-xjtsatMxit-satMxjt≤0, where we have used ∑i=1N∑j∈Vaijεiζi-ζj=1/2∑i=1N∑j∈Vaijεi-εjζi-ζj for any εi,ζi∈R,i=1,2,…,N, under the undirected and connected graph. Since we know sign(satMxi(t)-satMxj(t))=sign(xi(t)-xj(t)), there exists some constant ϵ1>0 such that 1≥satMxi(t)-satMxj(t)/xi(t)-xj(t)≥ϵ1. Then, we have (12)V˙t=-∑i=1N∑j∈Vaijxit-xjtsatMxit-satMxjt≤-ϵ1∑i=1N∑j∈Vaijxit-xjt2=-ϵ1λ2V1t, where λ2>0.
For a periodtd, if the discrete-time subsystem (6) is activated, the time shift of V(t) is(13)△Vt=∑i=1Nxi2t+1-∑i=1Nxi2t=-∑i=1N2xit-h∑j∈VaijsatMxit-satMxjt×h∑j∈VaijsatMxit-satMxjt=-h∑i=1N∑j∈Vaijxit-xjtsatMxit-satMxjt+h2∑i=1N∑j∈VaijsatMxjt-satMxit2. According to the Cauchy-Schwarz inequality, we have(14)△Vt=-h∑i=1N∑j∈Vaijxit-xjtsatMxit-satMxjt+h2∑i=1N∑j∈VaijaijsatMxjt-satMxit2≤-h∑i=1N∑j∈Vaijxit-xjtsatMxit-satMxjt+h2∑i=1N∑j∈Vaij∑j∈VaijsatMxjt-satMxit2=-h∑i=1N∑j∈VaijsatMxit-satMxjt×xit-xjt-h∑j∈VaijsatMxit-satMxjt≤0, where the fact is that the sign of xi(t)-xj(t)-h∑j∈VaijsatMxi(t)-satMxj(t) is equivalent to the sign of xi(t)-xj(t), since 0<h<1/maxi∈IN∑j∈Vaij. Furthermore, we define xi(t)-xj(t)-h∑j∈VaijsatMxi(t)-satMxj(t)=ϵ2xi(t)-xj(t), where 0<ϵ2<1. Then, we have (15)△Vt≤-ϵ2h∑i=1N∑j∈Vaijxit-xjtsatMxit-satMxjt≤-ϵ1ϵ2hλ2Vt. If we give a time period t=tc+td, where tc is the time for the continuous-time system (5) and td is the time for the discrete-time system (6), then we have (16)Vt≤e-ϵ1λ2tc1-ϵ1ϵ2hλ2tdV0=e-ϵ1λ2tce-tdln1/1-ϵ1ϵ2hλ2V0≤e-2γtV0, where γ=minϵ1λ2/2,ln1/1-ϵ1ϵ2hλ2/2. It follows that xi(t)≤e-γtxi(0),i∈IN. Furthermore, as for 1/N∑i=1Nxi(0)≤M, we know from (11) and (14) that V˙(t)=0 or △V(t)=0 only if xi(t)=xj(t) or satMxi(t)=satMxj(t),i,j∈IN. Thus, the switched MAS with (5) and (6) can achieve consensus if satMxi(t)=satMxj(t),i,j∈IN, can make xi(t)=xj(t),i,j∈IN. Note that satMxi(t)=satMxj(t),i,j∈IN, only when xi(t)≥M or xi(t)≤-M and xi(t)≤M with x1t=x2t=⋯=xN(t). Since 1/N∑i=1Nxi(0)≤M, then the condition that xi(t)≥M or xi(t)≤-M is not satisfied. Therefore, satMxi(t)=satMxj(t) only when xi(t)=xj(t),i,j∈IN is satisfied. This completes the proof.Remark 4.
With the Gershgorin disk theorem, the maximal eigenvalues ofL are satisfied: λN≤2maxi∈INdii=2maxi∈IN∑j∈Vaij. Thus, the sampling period can be replaced by 0<h<2/λN.Remark 5.
Note that iftc=0 (or td=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.In what follows, we consider the switched MAS with (5) and (6) on switched and undirected graphs. That is, the graph G(t) will be randomly switched among s distinct and finite topologies G(t)∈G1,G2,…,Gs, and G(t)=Gk if and only if the random switching signal σ(t)=k∈1,2,…,s. Then, the following results can be obtained.Theorem 6.
Assume graphGk is undirected and connected for each Gk∈G1,G2,…,Gs and the condition is 0<h<mink∈1,2,…,s2/λN(Lk), where the Laplacian matrix of graph Gk is Lk. Then, the switched MAS with (5) and (6) can solve the consensus problem if (17)1N∑i=1Nxi0≤M. Furthermore, the final consensus state is limt→∞xi(t)=1/N∑i=1Nxi(0).Proof.
Similar to Theorem3, it is proved that the average of all agents’ states 1/N∑i=1Nxi(t) is invariant, which implies the final consensus state is ξ=1/N∑i=1Nxi(0) if the consensus is reached.
In view of the proof of Theorem3, for a time period tc, if system (5) is switched on, we have (18)V˙t=-∑i=1N∑j∈Vaijtxit-xjtsatMxit-satMxjt≤-ϵ1λ2LtVt≤-ϵ1θVt, where θ=mink∈1,2,…,sλ2(Lk) and ϵ1 has been defined before.
For a time periodtd, if the discrete-time system (6) is switched on, the time shift of V(t) is shown as (19)△Vt≤-h∑i=1N∑j∈VaijtsatMxit-satMxjt×xit-xjt-h∑j∈VaijtsatMxit-satMxjt≤0, since 0<h<mink∈1,2,…,s2/λN(Lk). It is clear that V(t)=0 if and only if xi(t)=xj(t). Furthermore, we have (20)△Vt≤-h∑i=1N∑j∈VaijtsatMxit-satMxjt×xit-xjt-h∑j∈VaijtsatMxit-satMxjt≤-ϵ2h∑i=1N∑j∈Vaijtxit-xjtsatMxit-satMxjt≤-ϵ1ϵ2hθVt, where ϵ2 has been defined in Theorem 3. For any t=tc+td, we have (21)Vt≤e-ϵ1θtc1-ϵ1ϵ2hθtdV0=e-ϵ1θtce-tdln1/1-ϵ1ϵ2hθV0≤e-2ηtV0, where η=minϵ1θ/2,ln1/1-ϵ1ϵ2hθ/2. It follows that xi(t)≤e-ηtxi(0),i∈IN. Then, similar to the proof of Theorem 3, since 1/N∑i=1Nxi(0)≤M, we have V˙(t)=0 or △V(t)=0 only when xi(t)=xj(t),i,j∈IN. 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 Figure1(a). According to Theorem 3, the fact that sampling period condition is 0<h<0.4463 can be calculated. Let h=0.3 and M=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. Figure 2(a) is with the initial states x1(0)=-5,-7,-8,15,3T, and Figure 2(b) is with the initial states x2(0)=20,13,-10,-14,15T. Under the initial states x1(0), the absolute final value is 1/5∑i=15xi(0)=0.4≤M=2, which satisfies condition (9), and then the switched MAS can achieve consensus under arbitrarily given switching law. However, under the initial states x2(0), the absolute final value is 1/5∑i=15xi(0)=4.8≥M=2 in Figure 2(b), which does not satisfy condition (9), and thus the switched MAS cannot achieve consensus.Figure 1
The structure of switching topology with3 fixed and undirected graphs.
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Consensus of switched systems on undirected graph with fixed topologies.
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4.8Example 2.
Suppose that the switching undirected graph is given in Figure1, and the switching set G(t)=Gk∈Ga,Gb,Gc. The sampling period condition can be calculated as 0<h<min0.4463,0.5528,0.4796=0.4463. Let h=0.4 and M=5. In the top panel of Figure 3, we give the law of switching topologies, where the signals 1, 2, and 3 represent the graphs Ga,Gb, and Gc, 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 x3(0)=2,17,-15,-10,-5T. The figure shows that, with the condition (1/5)∑i=15xi3(0)=2.2<5 in Theorem 6, the switched MAS can achieve consensus.Figure 3
Consensus of switched systems on undirected graph with switching topologies.


## 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.



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