# Curvature- and Torsion-Based Consensus for the Markovian Switching Multiagent System with Communication of Noise Disturbance and Time Delay

Authors: Jinping Mou; Dongbing Tong; Xianyi Shao; Huafeng Ge; Yiling Lv
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper investigates the consensus problem for the distributed multiagent system (MAS), where the trajectory of each agent is displayed by curvature and torsion, and the communication behaviors among agents are influenced by time delay and corrupted by noises. According to the Frenet–Serret formulas, a class of consensus protocols is designed for all agents, and a closed-loop system is obtained. Based on the Lyapunov method, several consensus criteria are derived, where the consensus criteria are characterized by curvature functions and torsion functions. Finally, one example shows the reliability of the proposed methods.


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## Body

## 1. Introduction

In recent decades, the consensus problem of the multiagent system (MAS) has attracted the increasing attention of a large number of scholars, and the results have been widely used for tracking [1–5], formation control [6], distributed optimization monitoring [7, 8], and other purposes [9, 10].The consensus of MAS requires that the state of each node tends to be a common value. Generally, the consensus problem is to design the distributed protocol for MAS [11], and under the protocol, the states of all agents converge to a common value by the local communication. However, for the abrupt change of topology and stochastic disturbance, the local communication will encounter link failures in some real applications. It is well known that the switching topology is widely described by Markovian chain.Recently, many consensus results with Markovian switching have been investigated for MAS [12–23]. In [12], transition rates are considered to be not completely known; by the stochastic technique, sufficient conditions are expressed to solve the consensus problem for nonlinear MAS under the semi-Markovian switching topologies. In order to reduce the effect of communication noises, a time-varying gain in control protocol is utilized, and necessary and sufficient criteria are established to guarantee the mean square scaled consensus [13]. Considering the consensus control of grouped agents with linear or linearized nominal dynamics, a class of state-feedback controllers is proposed, and it shows that the controllers yield unbounded consensus region (see [14] and the references therein).Group-couple consensus with the Markovian switching system is a duration-significant research field. In [18], by employing linear transformations, couple-group consensus is considered for a class of discrete-time stochastic multiagent systems, and the performance of the reduced-order systems is provided to get constructive approaches for the fixed and the Markovian switching topologies. Under a designed switching signal set, the group consensus problem of the second-order MAS is studied in [24], where the switching topology obeys the Markovian chain, and by utilizing a state transformation method, two consensus sufficient criteria are established. Based on a system transformation, the problem of group-couple is converted to a L2 control issue, and sufficient conditions are obtained for MAS, where the topology obeys a Markovian switching with unknown transition probabilities [17].Event-triggered consensus problem with Markovian switching is another important issue on MAS, and it has attracted many scholars to study. In [25], in order to simplify information transmission, a distributed event-triggered mechanism is considered, and the proposed consensus protocol ensures MAS to achieve the consensus. By presenting an event-triggered control strategy, MAS achieves the fixed-time consensus [26–28].Fault-tolerant tracking is another research field of MSA. In [29], an adaptive fault-tolerant tracking protocol is proposed, and it ensures each follower agent to track a leader agent. . By considering the directed link failures or recoveries, a fault-tolerant control strategy with Markovian switching topology is provided for a nonlinear MAS, and some sufficient conditions of stochastic consensus are obtained, where MAS is in the presence of communication noises and actuator faults [30].Notice that most of the results mentioned above were obtained via the eigenvalue of matrix or the LMI method. In order to get the criteria of the consensus for MAS, some inherent relations will be neglected because of the inequalities to be utilized, such as curvature function and torsion function of each node when we consider the track problems. According to the theory of classic differential geometry, all tracks of agents are decided by their curvature and torsion entirely except for the difference of their locations. If all agents keep the same curvature function and torsion function, then they can move with the same track, and they can keep a certain distance among them. Recently, the consensus problem of MAS in the sense of curvature and torsion with fixed topology was studied in literature [31]. However, the results may not suitable to the switching system with stochastic disturbance. Motivated by this, in this paper, curvature- and torsion-based consensus problem with switching topology and noise disturbance is studied.The remainder of this paper is organized as follows. Section2 introduces some concepts. Section 3 proposes the consensus problem with Markovian switching. In Section 4, several consensus criteria are proposed for MAS. Section 5 provides a numerical example to show the reliability of the proposed methods. In Section 6, several conclusions are drawn.Notations. The main notations which will be used in this paper are listed in the following. Z+ is the positive integer set, Rn is the n- dimensional Euclidean space, C3 is the 3-times continuously differentiable, In is the n times n unit matrix, and 0 is the zero vector.


## 2. Preliminaries

LetV=v1,v2,…,vi,…,vn be the agent set, where i∈Z+, 1≤i≤n. Let A=aijn×n be a weighted symmetric adjacent matrix of all agents, where aij is associated with an edge which is denoted by i,j. aij takes value in 1 or 0, and aij=aji. aij=1 refers to the fact that there exists information flow between nodes vj and vi, or else aij=0. Let E=i,j|aij=1,i≠j be the edge set, where i,j is the edge between vi and vj. The neighbor set of vi is denoted by Ni, where Ni=vj|vi,vj∈E.Based onV, A, and E, the topology of the multiagent system (MAS) can be presented by a weighted graph G=V,E,A.LetGϱl=V,Eϱl,Aϱl be the interaction topology of dynamic agents, where(1)ϱt=ϱl,t∈t,t+a,a>0,and Gϱl refers to the time-varying topology which is driven by a random press ϱt, and ϱl∈T, where T=ϱ1,ϱ2,…,ϱi,…ϱm. The infinitesimal generator of the Markov chain ϱt,t≥0 is given by(2)Pϱt+a=ϱj|ϱi=λijh+oh,ifi≠j,1+λiih+oh,otherwise,where λij is the transition rate from state ϱi to ϱj; if ϱi≠ϱj, then λij≥0, ∑i≠jλij, and limh⟶0oh/h=0. ∀vi∈V, let(3)rit=xit,yit,zitT∈R3be the state of agent vi, where t∈R, and xit,yit,zit are the C3 functions; hence, the state of MAS is shown by(4)rt=r1tT,r2tT,…,ritT,…,rntTT.Denote(5)αit=r′tr′t,γit=r′t×r″tr′t×r″t,βit=γit×αit,where αit is the tangent unit vector, βit is the normal unit vector, and γit is the binormal unit vector, and they satisfy the following Frenet–Serret formulas:(6)αi′t=κitβit,βi′t=−κitαit+τitγit,γi′t=−τitβit,where κit and τit are the curvature function and torsion function of vi, respectively.As is known to all, the trajectory ofvi is decided entirely by the curvature function and the torsion function. Therefore, ∀vi and vj, if(7)κit=κjt,τit=τjt,then vi and vj have the same trajectory except for the difference of their locations.For convenience, we denoteτ^=maxτit|1≤i≤n, κ^=maxκit|1≤i≤n, τˇ=minτit|1≤i≤n, κˇ=minκit|1≤i≤n.Assumption 1.
We assume that(8)κi′t=0,t∈t,t+a,τi′t=0,t∈t,t+a.Consequently, based on the curvature function and torsion function, a consensus definition is proposed in the following.Definition 1.
∀vi, vj∈V, if(9)limt⟶∞Eκit−κjt=0,limt⟶∞Eτit−τjt=0,then MAS is the consensus in a broad sense of curvature and torsion (BCT consensus).Remark 1.
In literature [31], CT consensus in the sense of curvature and torsion refers to the fact that(10)limt⟶∞κit−κjt=0,limt⟶∞τit−τjt=0,and hence CT consensus is the special case of BCT consensus.Remark 2.
In the traditional literatures, e.g., see [8, 32], the consensus refers to the fact that∀vi,vj∈V,(11)limt⟶∞rit−rjt=0,and it follows that(12)limt⟶∞κit−κjt=0,limt⟶∞τit−τjt=0,and the state error between every two agents converges to zero vector. However, in some applications, we need all agents to keep nonzero distances while their trajectories possess the same shape.
According to the Definition1, every two agents can keep the nonzero distance. Meanwhile, all trajectories of agents converge to an interest curve.Remark 3.
In Definition1, if(13)limt⟶∞Eαit−αjt=0,limt⟶∞Eβit−βjt=0,limt⟶∞Eγit−γjt=0,based on equation (6), it follows that(14)βit⋅αi′t=κit,and it holds that(15)limt⟶∞Eκit−κjt=0.
Similarly, it gets that(16)limt⟶∞Eτit−τjt=0.


## 3. Problem Statement

In this section, the protocols are provided for tangent vector, normal vector, and binormal vectors firstly, and then the dynamics of tangent vectors, normal vectors, and binormal vectors are described for MAS.∀vi∈V, the dynamics of αit,βit,γit are described by(17)αi′t=κitβit+ui1t,βi′t=−κitαit+τitγit+ui2t,γi′t=−τitβit+ui3t,where ∀t∈s,s+a,κi′t=0,τi′t=0, ui1t, ui2t, and ui3t are the protocols of agent vi, and they are defined by(18)ui1t=κit−a∑j∈Niaijαjt−a−αit−a+σtwij−1di∑j∈Niaijκitβjt,ui2t=τit−a−κit−a∑j∈Niaijβjt−a−βit−a+σtwij+1di∑j∈Niaijκitαjt−τitγjt,ui3t=τit−a∑j∈Niaijγjt−a−γit−a+σtwij+1di∑j∈Niaijτitβjt,where wij is the white noise, σt≥0, and random press ϱt and wij are independent of each other.From (17) and (18), we obtain that(19)α′tβ′tγ′t=0Π120Π210Π230Π320αtβtγt−KϱlLϱl000−Tϱl−KϱlLϱl000TϱlLϱlαt−aβt−aγt−a+Λ11000Λ22000Λ33σtCW,where(20)α=α1Tt,α2Tt,…,αnTtT,β=β1Tt,β2Tt,…,βnTtT,γ=γ1Tt,γ2Tt,…,γnTtT,WiT=wi0,wi1,…,winT,W=W1T,…,WnTT.DenoteXt=αTt,βTt,γTtT, and we have(21)X′t=ΠtXt−ΣtXt−a+ΣtσtCW,where(22)Π=0Π120Π210Π230Π320,Σt=Σ1000Σ2000Σ3,Dϱι=diag1d1,1d2,…,1dn,Π12=DKϱlLϱl,Π21=−DϱlKϱlLϱl,Π23=DϱlTϱlLϱl,Π32=−DϱlTϱlLϱl,Σ1=KϱlLϱl,Σ2=−Kϱl−TϱlLϱl,Σ3=TLϱl,Kϱl=κ1t,κ2t,…,κnt,Tϱl=τ1t,τ2t,…,τnt,where C is a n2×n matrix, and(23)C=0a12⋯a1n0000⋯00⋯0a210⋯a2n0⋯0⋮⋮⋮⋮⋮⋮⋮⋮⋯⋮0000⋯00an1⋯0.In the following, theBCT consensus of system (21) will be studied.


## 4. The Main Results

In this section, the consensus problem is studied for system (21). Firstly, the state error of system (21) is discussed. Let x0t=α0t,β0t,γ0t be the aim state of the MAS, where(24)α0t=1n∑i=1nαi,β0t=1n∑i=1nβi,γ0t=1n∑i=1nγi.Denoteet=Xt−x0t⊗In+1=δ1tT,δ2tT,δ3tTT; then, it follows that(25)δ1t=αt−α0t,δ2t=βt−β0t,δ3t=γt−γ0t,and it holds that(26)et=Xt−x0t⊗In+1=I−PXt,where(27)P=P11000P22000P33,P11=P22=P33=1n1n×n.It follows that(28)e′t=I−PX′t=I−PΠXt−I−PΣtXt−a+I−PΣtσtCW.Notice that1n×n/nLϱι=0, and(29)∑j∈Niαjt−αit=∑j∈Niαjt−α0t−αit+α0t=∑j∈Niejt−eit.Hence,LϱιXt=Lϱιet, and system (28) can be shown by(30)e′t=I−PΠtet+I−PΣtet−a+I−PΣtCW,(31)det=fϱι,et,et−adt+gϱι,et,et−adBt,where(32)fϱι,et,et−a=I−PΠtet−I−PΣttet−a,gϱι,et,et−a=I−PΣtσtC.Lemma 1.
(see [33]). Let a′t≤a¯t>0 hold. Assume that there is a function V∈C2,1Rn×R+;R+ and positive constants p,c1,c2,c3,c4 such that c3>c4/1−a¯,(33)c1etp≤Vet,t,ϱι≤c2etp,LV≤et,t,ι≤−λ1etp+λ2et−ap,for all et,et−a∈Rn, t≥0, and ρι∈T. Then, equation (31) is asymptotically stable in pth moment.Lemma 2.
(see [33]). Let a′t≤a¯t>0 hold and p≥2. Assume that for each ι∈T, there is a pair of real numbers ςρι and σρι such that(34)etTfet,et−at+p−12get,et−at2≤ςριet2+σριet−a2,for all et,et−a,t∈Rn×Rn×R+. Assume that(35)ℬ≔−diagpςρ1,…,pςρm−Γis a nonsingular M-matrix, so q1,…,qmT:=ℬ−11⟶≫0. If(36)pqρισρι<1−a¯,ςρι∈T,then equation (31) is a asymptotically stable in pth moment.Then, we discuss the stability of system (31). Let the Lyapunov candidate function be presented as(37)Vt=12etTet,and it follows that(38)etTI−PΠtet=δ1tT,δ2tT,δ3tTI−PΠδ1tT,δ2tT,δ3tTT=−δ2tTI−P22Π12δ1t+δ1tTI−P11Π12δ2t−δ3tTI−P33Π23δ2t+δ2tTI−P22Π23δ3t=0,etTI−PΣtet−a=δ1tT,δ2tT,δ3tTI−PΣtδ1t−aT,δ2t−aT,δ3t−aTT=∑i=13δitTI−PiiΣitδit−a=∑i=1312δitTI−PiiΣitδit+δit−aTI−PiiΣitδit−a−δit−δit−aTI−PiiΣitδit−δit−a.To estimate the value ofetTI−PΣtet−a, we need the following conditions.and we also need the following assumptions.Assumption 2.
In this paper, we assume that(39)κ^≥τˇ≥0,Assumption 3.
Letδ1˜, δ2˜, δ3˜ satisfy the following inequalities:(40)δ1˜≤k1δ1^,δ2˜≥k2δ2,δ3˜≤k3δ3^,where δ˜i=δit−δit−a.Under Assumption3, it holds that(41)∑i=13δitTI−PiiΣitδit≤∑i=13I−PiiΣitδitTδit,etTI−PΣtet−a=δ1tT,δ2tT,δ3tTI−PΣtδ1t−aT,δ2t−aT,δ3t−aTT=∑i=13δitTI−PiiΣitδit−a=∑i=1312δitTI−PiiΣitδit+δit−aTI−PiiΣitδit−a−δit−δit−aTI−PiiΣitδit−δit−a.Notice thatλPiiKL=0; then, it follows that(42)λI−PiiKL=λKL,and we have(43)λ2κˇ≤λKL≤λMκ^,and it holds that(44)−δ1tTI−P1Σ1δ1t=−δ1tTKLδ1t≤−κˇλ2δ1tTδ1t,δ2tTI−P2Σ2δ2t=δ2tTKLδ2t−δ2tTTLδ2t≤κ^λM−τˇλ2δ2tTδ2t,−δ3tTI−P3Σ3δ3t=−δ3tTKLδ3t≤−τˇλ2δ3tTδ3t.Similarly, it follows that(45)−δ1t−aTI−P1Σ1δ1t−a≤−κˇλ2δ1t−aTδ1t−a,δ2t−aTI−P2Σ2δ2t−a≤κ^λMδ2t−aTδ2t−a−τˇλ2δ2t−aTδ2t−a=κ^λM−τˇλ2δ2t−aTδ2t−a,−δ3t−aTI−P3Σ3δ3t−a≤−τˇλ2δ3t−aTδ3t−a.Assumption 4.
We assume that(46)hitδiTtδit≤δ˜iTtδ˜it≤gitδiTtδit,where δ˜it=δit−δit−a, hit≥0 and git≥0.In addition,δ˜it,i=1,2,3 satisfy the following inequalities:(47)δ˜1tTI−P1Σ1δ˜1t=δ˜1tTKLδ˜1t≤κ^λMδ˜1tTδ˜1t≤gitκ^λMδ1tTδ1t,δ˜2tTI−P2Σ2δ˜2t=−δ˜2tTKLδ˜2t+δ˜2tTTLδ˜2t,≤−h2tκˇλ2δ2tTδ2t+g2tτ^λ2δ2tTδ2t=−κˇλ2h2t+τ^λMg2tδ2tTδ2t,δ˜3tTI−P3Σ3δ˜3t=δ˜3tTKLδ˜3t≤τ^λMg3tδ3tTδ3t.For convenience, denote(48)f1δ1=−κˇλ2+gitκ^λMδ1tTδ1t,f2δ1=κ^λM−h2tκˇλ2−τˇλ2+τ^λMg2tδ2tTδ2t,f3δ1=−τˇλ2+τ^λMg3tδ3tTδ3t.On the other hand, it holds that(49)traceI−PΣtCTI−PΣtCσt≤traceCTΣtTΣtCσt=2σt2∑i=1sdiκi2+τi2−κiτi≤2sσt2dMκ^2+τ^2−κˇτˇ,etTfϱι,et,et−a+12gϱι,et,et−a2≤etTI−PΠtet+etTI−PΣtet−a+traceI−PΣtCTI−PΣtCσt≤f1δ1+f2δ2+f3δ3+2nσt2dMκ^2+τ^2−κˇτˇ,and hence it gets the following result.Theorem 1.
IfGϱl is connected and if(50)g1t=τ^λMκ^λMg3t,g3t=κ^τ^−κˇλ2τ^λMh2t+g2t,σt2<ν+τˇλ2−κ^λM2ndMκ^2+τ^2−κˇτˇet−aTet−a,where ν is a real number, and ν<1, 2qισι≤1, then system (31) is asymptotically stable, where λ2 is the second largest eigenvalue of L, A−2=diagω1,ω2,…,ωm−Γ is a nonsingular M-matrix, and q1,q2,…,qmT:=A−11⟶. Therefore, system (31) is the BCT consensus.Proof.
Define the functionV:Rn×R+×S⟶R+ by(51)Vδ,ϱι,t=qι∑j=13Vj=qι∑j=13δjTδj,with c1=minι∈Tqι and c2=maxι∈Tqι. Applying the generalised It o^ formula to eςjtVj, it holds that(52)LVj≤2qιδjtTfϱι,δjt,δjt−a+12gϱι,δjt,δjt−a2+∑r=1mγιrqιδjtTδjt,and hence(53)LV≤2qιetTfϱι,et,et−a+12gϱι,et,et−a2+∑r=1mγιrqιetTet≤2qι−κˇλ2+gitκ^λMδ1tTδ1t−κˇλ2δ1t−aTδ1t−a+∑r=1mγιrqιδ1tTδ1t+2qικ^λM−h2tκˇλ2−τˇλ2+τ^λMg2tδ2tTδ2t+κ^λM−τˇλ2δ2t−aTδ2t−a+2nσt2dMκ^2+τ^2−κˇτˇ+∑r=1mγιrqιδ2tTδ2t−τˇλ2+τ^λMg3tδ3tTδ3t−τˇλ2δ3t−aTδ3t−a+∑r=1mγιrqιδ3tTδ3t≤−δ1tTδ1t−κˇλ2δ1t−aTδ1t−a−δ2tTδ2t+κ^λM−τˇλ2δ2t−aTδ2t−a−δ3tTδ3t−τˇλ2δ3t−aTδ3t−a+2nσt2dMκ^2+τ^2−κˇτˇ≤−etTet+κ^λM−τˇλ2et−aTet−a≤−etTet+νet−aTet−a,and according to Lemma 1, system (31) is asymptotically stable.
So, it holds that(54)limt⟶∞Eet=0,and that system (31) is the BCT consensus.Theorem 2.
IfG is connected and if(55)g1t=τ^λMκ^λMg3t,g3t=κ^τ^−κˇλ2τ^λMh2t+g2t,σt2<ν+τˇλ2−κ^λM2ndMκ^2+τ^2−κˇτˇet−aTet−a,where ν is a real number, and ν<1, νeaς1=1, then system (31) is asymptotically stable. Therefore, system (31) is the BCT consensus.Proof.
Since(56)λeςjtVj+eςjtLVj≤ςjeςjtVj+eςjtfjt−κˇλ2δ1t−aTδ1t−a+2nσt2c3,where c3=dMκ^2+τ^2−κˇτˇ, then(57)c1eςjtEet2≤c2Eξj02+E∫0teςjs−esTes+νes−aTes−ads,where(58)∫0teς1sνes−aTes−ads≤νeaς1∫−ateς1s−aνes−aTes−ads−a=νeaς1∫0teς1sesTesds+νeaς1∫−a0eς1sesTesds.
Since it holds thatνeaς1=1,(59)c1eςjtEet2≤c2Eξj02+νeaς1Eξ12,and ς1 satisfies that(60)−λ12κˇ+12eaς1λMk1κ^−λ1κˇ=0,and according to Lemma 2, system (31) is asymptotically stable.
So, it holds that(61)limt⟶∞Eet=0,and that system (31) is the BCT consensus.


## 5. A Numerical Example

To illustrate the effect of the methods in the main results, an example is proposed in the following.Example 1.
Consider a connected MAS (21) with three agents, and the topologies of (21) are shown in Figure 1, where V=v1,v2,v3. According to the different topologies, the related curvature functions and torsion functions are provided in Table 1.(62)Xt=r1t,r2t,r3tT,r10=17,0,0T,r20=50,0,0T,r30=20,0,0T.
The related matrices of equation (21) are listed below.(63)Π12ϱ1=0.1−0.10−0.10.2−0.10−0.250.25=−Π21ϱ1,Π23ϱ1=0.21−0.210−0.050.1−0.050−0.220.22=−Π32ϱ1,Π12ϱ2=0.20−0.200.33−0.33−0.31−0.310.62=−Π21ϱ2,Π23ϱ2=0.210−0.2100.22−0.22−0.11−0.110.22=−Π32ϱ2,Π12ϱ3=0.23−0.115−0.115−0.20.20−0.2500.25=−Π21ϱ3,Π23ϱ3=0.2−0.1−0.1−0.10.1−0−0.2100.21=−Π32ϱ3,Σ1ϱ1=0.1−0.10−0.20.40.20−0.250.25,Σ2ϱ1=0.01−0.0100.1−0.20.100.03−0.03,Σ3ϱ1=0.21−0.210−0.10.2−0.10−0.220.22,Σ1ϱ2=0.20−0.200.33−0.33−0.31−0.310.62,Σ2ϱ2=0.010−0.010−0.110.110.110.11−0.22,Σ3ϱ2=0.210−0.2100.22−0.22−0.2−0.20.4,Σ1ϱ3=0.46−0.23−0.23−0.20.20−0.2500.25,Σ2ϱ3=−0.060.030.030.1−0.100.040−0.04,Σ3ϱ3=0.4−0.2−0.2−0.10.10−0.2100.21.
From the following Laplacian matrices:(64)Lϱ1=1−10−12−10−11,Lϱ2=10−101−1−1−12,Lϱ3=2−1−1−110−101,it obtains that λ2=1,λM=3. Suppose that ν=0.8, h2t=0.5,(65)σt,ϱ12<0.2et−aTet−a,σt,ϱ22<172et−aTet−a,σt,ϱ32<0.12et−aTet−a,g1t,ϱ=13,g2t,ϱ=κ^6τ^,g3t,ϱ=κ^τ^,Γ=−2111−2111−2.
According to Theorem1, under the control of protocol (17), MAS (21) achieves the BCT consensus (see the simulation results of Figures 2–4).
Figures2–4 show that under protocol (18), every component coordinate of vectors αit,βit,γit converges to a common value, respectively, curvature functions and torsion functions converge to a value, respectively, and all agents converge to the BCT consensus. In some applications, the protocol ensures that all agents keep the same trajectories; in the meanwhile, agents can also keep nonzero distances.Figure 1
The topologies of MAS (21).Table 1
The values of curvature functions and torsion functions with different topologies.
κ and τ values of κ and τtopologiesϱ1ϱ2ϱ3κ1,τ1(0.1, 0.21)(0.2, 0.1)(0.25, 0.23)κ2,τ2(0.2, 0.21)(0.33, 0.22)(0.31, 0.2)κ3,τ3(0.23, 0.2)(0.2, 0.1)(0.25, 0.21)Figure 2
Component coordinates of vectorsα1,α2, and α3 converging to a same value.Figure 3
Component coordinates of vectorsβ1,β2, and β3 converging to a same value.Figure 4
Component coordinates of vectorsγ1,γ2, and γ3 converging to a same value.


## 6. Conclusion

In this paper, we have studied the consensus problem in the sense of torsion and curvature for stochastic MAS with Markovian switching topology and time delay. A kind of distributed consensus protocol has been designed for MAS, and we have gotten the MAS which is updating upon torsion function and curvature function, where the protocol is based on the Frenet–Serret formulas. By employing the Lyapunov functional method and the stochastic approaching, several consensus criteria have been obtained, the results have shown that the protocol can ensure the system achieves the consensus, and the criteria have been analysed and illustrated by simulations. However, some challenging problems have not been considered in this paper, such as MAS in the small-world framework or systems with unknown partial Markovian transformation rate, and the related consensus problems will be studied by us in the future.



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*Source: 9028591-2021-03-25.xml*