# Stabilization of Networked Distributed Systems with Partial and Event-Based Couplings

Authors: Sufang Zhang; Wei Wang; Chi Huang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

The stabilization problem of networked distributed systems with partial and event-based couplings is investigated. The channels, which are used to transmit different levels of information of agents, are considered. The channel matrix is introduced to indicate the work state of the channels. An event condition is designed for each channel to govern the sampling instants of the channel. Since the event conditions are separately given for different channels, the sampling instants of channels are mutually independent. To stabilize the system, the state feedback controllers are implemented in the system. The control signals also suffer from the two communication constraints. The sufficient conditions in terms of linear matrix equalities are proposed to ensure the stabilization of the controlled system. Finally, a numerical example is given to demonstrate the advantage of our results.


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## Body

## 1. Introduction

Recent years have witnessed a thriving research activity on how to assemble and coordinate networked distributed systems (NDSs) into a coherent whole to perform a common task [1]. NDSs have obvious advantages in practice, such as energy saving, easy installation, and higher reliability [2–6]. Thus, studying stabilization of NDSs is of theoretical and practical importance. To realize the stabilization, a control strategy is needed. However, due to the absence of central data fusion, the classical centralized control scheme is not feasible for NDSs. Accordingly, the cooperative control strategy is a preferred choice. Since an NDS consists of a large number of agents, it is impossible and unnecessary to control every agent. An effective approach is to implement controllers for a fraction of the NDSs to stabilize the whole system, which is referred to as the pinning stabilization problem [7, 8].To achieve stabilization, the communication in NDSs plays a crucial role. However, due to physical and environmental limitations, communication constraints, such as time delays [9–11] and noise [12, 13], are unavoidable. In fact, incomplete information is universal. As each agent of an NDS has multiple levels of information, the coupling has to be split into multiple channels to transmit the corresponding levels of information. Due to the physical limitatioins, only some parts of the channels can transmit information successfully, which brings the partial-coupling problem. Such a phenomenon can be observed in many real systems. For example, in brain networks, only 5% excitatory synapses sent from a cortical area can be received by another connected cortical area [14]; in sensor networks, the information packet of a target may be partly lost during communication between sensors [5]. Thus, it is highly desirable to analyze NDSs with partial couplings.The sampling issue has received intense attention, ever since the rapid development of digital technology and intelligent equipment. A traditional sampling protocol is time-based sampling technique [15–17]. Recently, the event-based sampling has been investigated as an alternative to time-based sampling. The distinct feature of event-based sampling is the real-time scheduling algorithm. The information is sampled only when a certain event occurs, for example, when the system state exceeds a predefined threshold. The advantage of the event-based sampling is the capability of fast reacting to sudden events and, therefore, being more efficient [18, 19]. In an NDS, the event conditions are individually designed for each agent. Thus, the agents are sampled at mutually independent instants. This means that the event-based sampling scheme does not require a common sampling schedule, which makes it applicable for a system with large size. For the single system, an event-triggered method was designed in [20], in which the lower bound of two successive sampling instants was given; in [21, 22], the event-triggered controller was designed for networked control systems with transmission delay. In [23], the event-based control was used in multiagent system drive the agents to average consensus. However, until now, few results have been given for the NDS with partial and event-based couplings. The difficulty of this problem is threefold. Firstly, two communication constraints are considered, both of which make less information available for communication. Secondly, each agent samples information separately; however, they should cooperatively converge. Thus, how to design the event conditions of the agents? Finally, the stabilization conditions should be given to guarantee the stabilization of the NDS.In this paper, we focus on the stabilization problem of NDSs with partial and event-based couplings. By designing an event condition for each agent, an event-based sampling scheme is proposed for NDSs. Due to the constraint of partial information transmission, the channels are considered in the event condition. Thus, for different channels of one agent, the sampling instants are distinct. The sampled data are used for the communication among agents and building the feedback controllers. The sufficient conditions are given to ensure the stabilization of the controlled NDS with both communication constraints. Finally, a numerical example is given to demonstrate the advantage of our results.Notations. The standard notations will be used in this paper. Throughout this paper, R denotes the set of real numbers. Rn denotes the n dimensional Euclidean space. Rn×n are the set of n×n real matrices. In∈Rn×n is the identity matrix. For real symmetric matrices X and Y, the notation X≤Y (resp., X<Y) means that the matrix X-Y is negative semidefinite (resp., negative definite). · denotes the Euclidean norm for vector or the spectral norm of matrix. diag{⋯} denotes a diagonal matrix. The superscripts “⊤” and “-1” represent the matrix transposition and matrix inverse. ∗ in a matrix represents the elements below the main diagonal of a symmetric matrix. Matrices, if not explicitly stated, are assumed to have compatible dimensions.


## 2. Problem Formulation and Preliminaries

An NDS withN agents can be described by the following:(1)x˙it=Axit+ui1t,i∈1,2,…,N,where xi(t)=[xi1(t),xi2(t),…,xin(t)]∈Rn is the state of the ith system; A=[aij]n×n is the system matrix. Assuming that each agent can only receive the information of its neighbors, the coupling control ui1(t) can be constructed as follows:(2)ui1(t)=α∑j=1N‍gijRij(xj(t)-xi(t)),where α is the coupling strength; G=[gij]N×N is the Laplacian matrix representing the coupling structure of the NDS. The elements of G are defined as follows: if there is a connection from the jth agent to ith agent, then gij>0; otherwise, gij=0, and the diffusive coupling condition gii=∑j=1,j≠iN‍gij is satisfied. Rij, named aschannel matrix, is a diagonal matrix with the diagonal element rijk=0 or 1 (k=1,2,…,n).Remark 1.
Since the state of each agent consists ofn levels of information, the couplings in the NDS have to be divided into n channels to transmit the corresponding levels of information. Due to practical constraints, only part of the n channels can work normally. The diagonal element rijk (k=1,2,…,n) of the channel matrix Rij is employed to indicate the activity of the kth channel connecting agents j and i. Specifically, Rijxj=[rij1xj1,…,rijnxjn]⊤. When rijk=1, rijkxjk=xjk, the kth level of state of agent j can be transmitted to agent i (i.e., the kth channel of the connection isactive); otherwise, when rijk=0, rijkxjk=0, the kth level of state of xj is lost (i.e., the kth channel of the connection fails to transmit the information).In the coupling control (2), each agent can receive the real-time information of its neighboring agents. However, the real-time information will increase the burden of the communication media. More importantly, it is unnecessary. For the sake of energy saving, the event-based control mechanism has been recently proposed as an effective alternative to the more conventional execution of control tasks. In this paper, the communication of agents will be carried out in an event-based manner; that is, the state of the agent will be sampled, if a given event is triggered.To realize the event-based sampling, an event condition is designed for each agent. When the event condition is violated, the agent will sample its information and send it to its neighbors. Considering that the information is transmitted through channels, the event condition is given as follows:(3)xiktmik-xiktmik+lh≤γikxiktmik+lh,hhhhhhhhi∈{1,2,…,N},k∈{1,2,…,n},where l=1,2,…; γik is a positive scalar; h>0 is the sampling period; tmik is the latest sampling instant of the kth level of information of the ith agent. Thus, the next event-triggered instant is the time when event condition (3) is violated; that is,(4)tm+1ik=tmik+hinf⁡l∈N:xik(tmik)-xik(tmik+lh)hhhhhhhhhhhhhhh>γikxi(tmik+lh).Without loss of generality, t0ik=0 is the initial sampling time, for all i and k.Letx^ik(t)=xik(tmik), for t∈[tmik,tm+1ik) and x^i(t)=[x^i1(t),x^i2(t),…,x^in(t)]. Thus, NDS (1) with partial and event-based couplings can be described as(5)x˙i(t)=Axi(t)+α∑j=1,j≠iN‍gijRij(x^j(t)-x^i(t)).Remark 2.
Due to the traffic jam and physical characteristics of transmission media, communication constraints, such as transmission delay [24–26], data packet dropout [27], and noise [28], commonly happen in real systems. In this paper, two kinds of communication constraints including partial information transmission of system and event-based sampled data are simultaneously considered in the couplings of NDS (5). Although both constraints cause information loss, their mechanisms are different. The event-based sampled data makes the real-time information available at the instants when event condition (3) is violated. The constraint of partial information transmission implies that the information sent at every instant is a lack of integrity. When we consider these two communication constraints simultaneously in NDSs, only part of the sampled information can be used for coupling control. Thus, the stabilization of the NDS is much harder to be realized.Remark 3.
Event condition (3) governs the sampling operation of NDS (5). When a subsystem is diverging, that is, the left hand side of the event condition becomes large, the subsystem has to update its state by sampling. In other words, the sampling happens only when it is needed. Thus, the event-based sampling scheme is more flexible and effective. It is able to realize fast reaction to emergency and avoid redundant samplings. Besides energy saving, another advantage of event-based sampling scheme is feasibility for large-scale systems. The sampling instants are independently decided by the event conditions of different subsystems. Thus, the common sampling schedule is unnecessary, which is hardly carried out in a system with large number of components. These two advantages make the event-based sampling scheme more applicable in engineering.In order to stabilize the NDS, the state feedback controllers will be implemented. Considering that it is difficult and unnecessary to install the controller for every agent, we only choose a small fraction of agents to be controlled. In addition, we also assume that the control signal suffers from the two communication constraints. Thus, the pinning controller of agenti can be constructed as follows:(6)ui2(t)=-diHix^i(t),where di≥0 is the control strength. In particular, when di=0, it means that the ith agent will not be controlled. Hi=diag{hi1,hi2,…,hin} with hik=1 or 0, which indicates that the kth level of the event-based sampled information of x^(t) can be or cannot be sent from the controller.Combining (5) and (6), the controlled NDS with partial and event-based couplings can be described by following equation:(7)x˙i(t)=Axi(t)+α∑j=1,j≠iN‍gijRij(x^j(t)-x^i(t))-diHix^i(t).Note that, fort∈[tmik+lh,tmik+lh+h) (l≥0, m≥0, i=1,2,…,N and k=1,2,…,n),(8)x^jkt-x^ikt=xjk(tm′jk)-xik(tmik)=xjk(tm′jk)-xjk(tmik+lh)-xik(tmik)+xik(tmik+lh)+xjk(tmik+lh)-xik(tmik+lh)=ejk(tmik+lh)-eik(tmik+lh)+xjk(tmik+lh)-xik(tmik+lh),where eik(tmik+lh)=x^ik(tmik+lh)-xik(tmik+lh), tm′jk=max⁡t∣t∈tmjk,m=0,1,…,t≤tmik+lh, and xjk(tm′jk)=xjk(tmjk+lh). Let ei(t)=[ei1(t),ei2(t),…,ein(t)]⊤. Thus, for t∈[mh,(m+1)h), (7) can be written as(9)x˙it=Axi(t)+α∑j=1,j≠iN‍gijRij(ej(mh)+xj(mh))-diHi(xi(mh)+ei(mh)).Let Cij=gijRij, for i,j=1,2,…,N, i≠j and Cii=-∑j=1,j≠iN‍Cij. Thus, Cij are diagonal matrices with diagonal elements cij1,cij2,…,cijn (cijk=gij·rijk). For each k=1,2,…,n, it can be followed from (9) that(10)x˙ikt=∑j=1n‍akjxij(t)+α∑j=1N‍cijk(ejk(mh)+xjk(mh))-dihik(ejk(mh)+xjk(mh)).Let δxk(t)=[x1k(t),x2k(t),…,xNk(t)]⊤ and δek(t)=[e1k(t),e2k(t),…,eNk(t)]⊤. From (10) we can get(11)δxk˙(t)=∑j=1n‍akjδxj(t)+(αCk-Dk)(δek(mh)+δxk(mh)),where Ck=[cijk]N×N and Dk=diag{d1h1k,d2h2k,…,dNhNk}.The following definition and lemma are needed for the derivation of our main results in this paper.Definition 4.
NDS (7) with partial and event-based couplings is said to achieve globally exponential stabilization, if there exist M>0, ε>0 such that xit2<Me-εt is satisfied with any initial states xi(0) for ∀i∈{1,2,…,N}.The purpose of this paper is to propose a set of sufficient conditions for controlled NDS (7) with partial and event-triggered communication to ensure the globally exponential stabilization.Lemma 5 (see [29]).
For any real vectorsa, b and scalar ɛ>0, one has (12)a⊤b+b⊤a≤εa⊤a+ε-1b⊤b.


## 3. Main Results

In this section, the stabilization conditions will be derived for controlled NDS (7) with partial and event-based couplings.Theorem 6.
For anyk=1,2,…,n, let Γk=diag{γ1k2,γ2k2,…,γNk2}. NDS (7) with partial and event-based couplings can be globally exponentially stabilized, if there exist positive scalars ε1k, ε2k and matrices Pk>0, Qk=diag{q1k,q2k,…,qNk}>0, Wk=W1kW2kW4k∗W3k0∗∗W5k>0, U1k, U2k, U3k, U4k, V1k, V2k, Xk, X1k, such that the following LMIs are satisfied:(13)Pk+hXk+Xk⊤2+W1k-hXk+hX1k-W1kW2kW4k∗-hX1k-hX1k⊤+hXk+Xk⊤2+W1k-W2k-W4k∗∗W3k0∗∗∗W5k>0,(14)Φ11kΦ12kΦ13kΦ14khU1k⊤nV1k⊤0∗Φ22kΦ23kΦ24khU2k⊤00∗∗Φ33kΦ34khU3k⊤0nV2k⊤∗∗∗Φ44khU4k⊤00∗∗∗∗-hW1k00∗∗∗∗∗-nε1kIN0∗∗∗∗∗∗-nε2kIN<0,(15)Φ11kΦ12kΠ13kΦ14knV1k⊤0∗Π22kΠ23kΦ24k00∗∗Π33kΠ34k0nV2k⊤∗∗∗Π44k00∗∗∗∗-nε1kIN0∗∗∗∗-nε2kIN<0,where(16)Φ11k=-Xk+Xk⊤2-U1k-U1k⊤+∑j=1n‍(ε1j+ε2j)ajk2In,Φ12k=Xk-X1k-W2k+U1k⊤-U2k+V1k⊤(αCk-Dk),Φ13k=Pk-U3k-V1k⊤,Φ14k=-W4k-U4k+V1k⊤(αCk-Dk),Φ22k=X1k+X1k⊤-Xk+Xk⊤2+W2k+W2k⊤+U2k+U2k⊤+QkΓk-hW3k,Φ23k=U3k+(αCk⊤-Dk)V2k,Φ24k=W4k+U4k,Φ33k=-V2k-V2k⊤,Φ34k=V2k⊤(αCk-Dk),Φ44k=-Qk-hW5k,Π13k=hXk+Xk⊤2+Pk-U3k-V1k⊤,Π22k=X1k+X1k⊤-Xk+Xk⊤2+W2k+W2k⊤+U2k+U2k⊤+QkΓk+hW3k,Π23k=hW2k⊤+hX1k⊤-hXk⊤+U3k+(αCk⊤-Dk)V2k,Π33k=hW1k-V2k-V2k⊤,Π34k=hW4k+V2k⊤(αCk-Dk),Π44k=-Qk+hW5k.Proof.
Fort∈[mh,(m+1)h), define the Lyapunov functional V(t)=∑k=1n‍Vk(t), where(17)Vkt=δxk⊤(t)Pkδxk(t)+ς(t)∫mht‍δx˙k(s)δxk(mh)δek(mh)⊤Wkδx˙k(s)δxk(mh)δek(mh)ds+ς(t)δxk(t)δxk(mh)⊤Xkδxk(t)δxk(mh),and ς(t)=(m+1)h-t, Xk=(Xk+Xk⊤)/2-Xk+X1k∗-X1k-X1k⊤+(Xk+Xk⊤)/2.
Before proceeding our proof, it should be pointed out thatV(t) is well defined. Based on Wk>0 and Schur complement [30], we have that (18)W^k≜W1k-W2kW3k-1W2k⊤-W4kW5k-1W4k⊤.Thus, it follows that(19)Wk-W^k00∗00∗∗0=W2kW3k-1W2k⊤+W4kW5k-1W4k⊤W2kW4k∗W3k0∗∗W5k≥0,which implies (20)Wk≥W^k00∗00∗∗0.Based on the above inequality, it can be got that(21)δx˙k(s)δxk(mh)δek(mh)⊤Wkδx˙k(s)δxk(mh)δek(mh)≥δx˙k⊤(s)W^kδx˙k(s).Thus, it follows from (17) and (21) that(22)Vkt≥δxk⊤(t)Pkδxk(t)+ςt∫mht‍δx˙k⊤sW^kδx˙ksds+ς(t)δxk(t)δxk(mh)⊤Xkδxk(t)δxk(mh).By applying the Jensen inequality [31], we have(23)∫mht‍δx˙k⊤(s)W^kδx˙k(s)ds≥1τ(t)∫mht‍δx˙k⊤(s)dsW^k∫mht‍δx˙k(s)ds≥1hδxkt-δxkmh⊤W^k[δxk(t)-δxk(mh)],where τ(t)=t-mh. From (22) and (23), it can be obtained that (24)Vk(t)≥δxk(t)δxk(mh)⊤Hkδxk(t)δxk(mh),where Hk=(ς(t)/h)[Pk+hXk+W^k]+(τ(t)/h)Pk, Pk=Pk0∗0 and W^k=W^k-W^k∗W^k. Based on the Schur complement [30], it follows from (13) that Pk+hXk+W^k>0. Thus, we can always find a positive scalar β<1, such that Pk+hXk+W^k>βPk, which further means that(25)Vt=∑k=1n‍Vk(t)>∑k=1n‍βδxk(t)δxk(mh)⊤Pkδxk(t)δxk(mh)=β∑i=1N‍xi(t)2.Therefore, V(t) defined in (17) is a valid Lyapunov functional.
The derivative ofVk(t) can be got that(26)V˙kt=2δx˙k(t)Pkδxk(t)-∫mht‍δx˙k(s)δxk(mh)δek(mh)⊤W~kδx˙k(s)δxk(mh)δek(mh)ds+ς(t)δx˙k(t)δxk(mh)δek(mh)⊤Wkδx˙k(t)δxk(mh)δek(mh)-δxktδxkmh⊤Xkδxktδxkmh+2ςtδx˙kt0⊤Xkδxktδxkmh-χ1∫mht‍δx˙k(s)2ds,where W~k=Wk-diag{χ1IN,0,0} and χ1 is a positive scalar such that W~k>0. By employing the Jensen inequality [31], it follows that(27)∫mht‍δx˙k(s)δxk(mh)δek(mh)⊤W^kδx˙k(s)δxk(mh)δek(mh)ds≥τ(t)ϕk⊤(t)(W1k-χ1IN)ϕk(t)+τ(t)δxk⊤(mh)W3kδxk(mh)+τ(t)δek⊤(mh)W5kδek(mh)+2δxkt-δxkmh⊤W2kδxk(mh)+2δxkt+δxkmh⊤W4kδek(mh),where ϕ(t)=(1/τ(t))∫mht‍δx˙k(s)ds.
According to the Newton-Leibnitz formula, it can be obtained that(28)0=2δxk⊤(t)U1k⊤+δxk⊤(mh)U2k⊤+δxk˙(t)U3k⊤δek⊤(mh)U4k⊤hhδxk⊤(t)U1k⊤+δxk⊤(mh)U2k⊤+δxk˙(t)U3k⊤+δek⊤(mh)U4k⊤·[-δxk(t)+δxk(mh)+τ(t)ϕk(t)].From (11), one can have that(29)0=2[δxk⊤(t)V1k⊤+δx˙k(t)V2k⊤]·-δx˙k(t)+∑j=1n‍akjδxj(t)hh∑j=1n‍+αCk-Dkδekmh+δxkmh.By employing Lemma 5, it follows that(30)2δxk⊤(t)V1k⊤∑j=1n‍akjδxj(t)≤nε1k-1δxk⊤tV1k⊤V1kδxkt+ε1k∑j=1n‍akj2δxj⊤tδxjt,2δx˙k⊤(t)V2k⊤∑j=1n‍akjδxj(t)≤nε2k-1δx˙k⊤(t)V2k⊤V2kδx˙k(t)+ε2k∑j=1n‍akj2δxj⊤(t)δxj(t).From event condition (3), it can be got that |eik(mh)|≤γik|xik(mh)|, which is equivalent to qik|eik(mh)|≤qikγik|xik(mh)|. Thus, we have that(31)δek⊤(mh)Qkδek(mh)≤δxk⊤(mh)QkΓkδxk(mh).Combining (26)–(31) yields that(32)V˙t=∑k=1nηk⊤(t)ς(t)hΨ¯k+τ(t)hΨkηk(t)∫mht‍δx˙k(s)2dshhhhhh-χ1∫mht‍δx˙k(s)2ds,where ηkt=δxktδxkmhδx˙ktδekmhϕkt⊤ and Ψ¯k=Ψk′0∗0,(33)Ψk=Φ11k+nε1k-1V1k⊤V1kΦ12kΦ13kΦ14khU1k⊤∗Φ22kΦ23kΦ24khU2k⊤∗∗Φ33k+nε2k-1V2k⊤V2kΦ34khU3k⊤∗∗∗Φ44khU4k⊤∗∗∗∗-hW1k+hχ1IN,Ψk′=Φ11k+nε1k-1V1k⊤V1kΦ12kΠ13kΦ14k∗Π22kΦ23kΦ24k∗∗Π33k+nε2k-1V2k⊤V2kΠ34k∗∗∗Π44k.By using the Schur complement [30], it follows from (14) and (15) that Ψkχ1=0<0 and Ψk′<0. Furthermore, we can always choose sufficiently small scalars χ1, χ2, χ3, and χ4 to ensure Ψk+diag{χ2IN,χ3IN,0,χ4IN,0}<0 and Ψk′+diag{χ2IN,χ3IN,0,χ4IN}<0. It follows from (32) that(34)V˙t<∑k=1n-χ2δxk(t)2-χ3δxk(mh)2∫mht‍hhhh-χ4δek(mh)2-χ1∫mht‍δx˙k(s)2ds.From (17), we have that(35)δxk⊤(t)Pkδxk(t)+ς(t)δxk(t)δxk(mh)⊤Xkδxk(t)δxk(mh)=δxk(t)δxk(mh)⊤ς(t)h[Pk+hXk]+τ(t)hPk·δxktδxkmh≤δxktδxkmh⊤·ςthPk+hXk+Uk+τthPkδxk(t)δxk(mh),where Uk=W1k-W1kW1k≥0. From (13), we have that there exists a positive scalar ρ such that(36)Pk+hXk+Uk<ρ1I2N,Pk<ρ1IN.Furthermore, a positive number ρ2 can be chosen such that(37)ς(t)∫mht‍δx˙k(s)δxk(mh)δek(mh)⊤Wkδx˙k(s)δxk(mh)δek(mh)ds≤ρ2∫mht‍δx˙k(s)2ds+ρ2δxk(mh)2+ρ2δek(mh)2.Thus, from (35) to (37), it follows that(38)Vt≤∑k=1nρ1δxk(t)2+(ρ1+ρ2)δxk(mh)2∫mht‍hhihhh+ρ2δek(mh)2+ρ2∫mht‍δx˙k(s)2ds.Let ε>0 such that(39)ερ1-χ2≤0,ε(ρ1+ρ2)-χ3≤0,ερ2-χ4≤0,ερ2-χ1≤0.According to (34), (38), and (39), we can get V˙(t)+εV(t)≤0∀t∈[mh,(m+1)h). Thus, V(t)≤e-ε(t-mh)V(mh). By mathematical induction, it can be concluded that V(t)≤e-εtV(0), ∀t>0. From (25), we yield that (40)xi(t)2≤∑i=1N‍xi(t)2≤β-1V(t)≤β-1V(0)e-εt,hhhhhhhhhhhhhhhhhhhhhhhhi∈{1,2,…,N}.Hence, from Definition 4, controlled NDS (1) with partial and event-based couplings can achieve globally exponentially stabilization under event condition (3) and pinning controllers (6). This completes the proof.When all the channel matricesRij are identity matrices, that is, Rij=In (i,j=1,2,…,N), the communication constraint of partial information transmission is removed. Furthermore, we assume that if an agent is controlled, every level of information of the controller can be well transmitted by the controller; that is, Hi (i=1,2,…,N) in (6) are also identity matrices. Overall, the controlled NDS with event-based couplings can be constructed by following differential equation:(41)x˙i(t)=Axi(t)+α∑j=1,j≠iN‍gij(x^j(t)-x^i(t))-dix^i(t).The main difference between NDS (7) and (41) is that the channels are not considered in (41). Thus, the event condition of each system can be described as(42)xi(tmi)-xi(tmi+lh)≤γixi(tmi+lh),hhhhhhhhhhhhhhhhhhhhi∈{1,2,…,N},where l=1,2,…, γi>0. Similarly, the next event-triggered instant is tm+1i=tmi+hinf⁡{l∈N:xi(tmi)-xi(tmi+lh)>γixi(tmi+lh)}. We also assume that t0i=0 for each agent i.Based on Theorem6, the stabilization conditions of NDS (41) with event condition (42) can be easily obtained as follows.Corollary 7.
LetΓ=diag{γ12,γ22,…,γN2} and D=diag{d1,d2,…,dN}. NDS (41) with event-based couplings can be exponentially stabilized, if there exist positive scalars ε1, ε2 and matrices P>0, Q=diag{q1,q2,…,qN}>0, W=W1W2W4∗W30∗∗W5>0, U1, U2, U3, U4, V1, V2, X, X1, such that the following LMIs are satisfied:(43)Pk+hX+X⊤2+W1-hX+hX1-W1W2W4∗-hX1-hX1⊤+hX+X⊤2+W1-W2-W4∗∗W30∗∗∗W5>0,Φ11Φ12Φ13Φ14hU1⊤nV1⊤0∗Φ22Φ23Φ24hU2⊤00∗∗Φ33Φ34hU3⊤0nV2⊤∗∗∗Φ44hU4⊤00∗∗∗∗-hW100∗∗∗∗∗-nε1IN0∗∗∗∗∗∗-nε2IN<0,Φ11Φ12Π13Φ14nV1⊤0∗Π22Π23Φ2400∗∗Π33Π340nV2⊤∗∗∗Π4400∗∗∗∗-nε1IN0∗∗∗∗-nε2IN<0,where(44)Φ11=-X+X⊤2-U1-U1⊤+∑j=1n‍(ε1j+ε2j)aj2,Φ12=X-X1-W2+U1⊤-U2+V1⊤(αG-D),Φ13=P-U3-V1⊤,Φ14=-W4-U4+V1⊤(αG-D),Φ22=X1+X1⊤-X+X⊤2+W2+W2⊤+U2+U2⊤+QΓ-hW3,Φ23=U3+(αG⊤-D)V2,Φ24=W4+U4,Φ33=-V2-V2⊤,Φ34=V2⊤(αG-D),Φ44=-Q-hW5,Π13=hX+X⊤2+P-U3-V1⊤,Π22=X1+X1⊤-X+X⊤2+W2+W2⊤+U2+U2⊤+QΓ+hW3,Π23=hW2⊤+hX1⊤-hXk⊤+U3+(αG⊤-D)V2,Π33=hW1-V2-V2⊤,Π34=hW4+V2⊤(αG-D),Π44=-Q+hW5.


## 4. Numerical Example

In this section, a numerical example will be given to demonstrate the effectiveness of our main results.An NDS with 4 agents is constructed (N=4), the topological structure of which is depicted in Figure 1. The Laplacian matrix of the NDS is given as G=-31111-21001-21101-2. The other parameters of the NDS are α=0.35 and A=0.12-0.08-0.060.12. Thus, each agent of the NDS has 2 levels of information (n=2). Due to the constraint of the partial information transmission, the couplings have to be divided into 2 levels of channels to transmit the information. For each coupling of the NDS, a channel matrix is given to reflect the work state of the channels of the coupling. The channel matrices of the couplings are listed as follows:(45)R12=diag{0,1},R13=diag{0,1},R14=diag{1,0},R21=diag{1,0},R23=diag{0,1},R32=diag{1,0},R34=diag{0,1},R41=diag{0,1},R43=diag{1,0}.Thus, the Laplacian matrices of the channels of the 2 levels can be, respectively, built: (46)C1=-10011-10001-10001-1,C2=-21100-11000-11100-1.As discussed above, the event-based sampling scheme is used in the communication among agents of the NDS and the control signals. We assume that, in event condition (3), the sampling period h=0.1 and γik=0.15, for i=1,2,3,4 and k=1,2. Let the coupling strength α=0.35. Without controllers, the constructed NDS is not stable. The trajectories of the uncontrolled NDS are shown in Figure 2, where i=1,2,3,4.Figure 1
The topological structure of the NDS with 4 subsystems.Figure 2
Without controllers, the trajectories of the NDS are divergent.To achieve stabilization, controllers are distributed, implemented in the NDS with the control strengths:d1=1, d2=1, d3=1, and d4=0; that is, except the 4th agent, the other agents are controlled. Furthermore, for the controlled agents, only part of the control signals can be well received. Let H1=diag{1,0}, H2=diag{0,1}, and H3=diag{1,1}. Thus, D1=diag{1,0,1,0} and D2=diag{0,1,1,0}. Based on the above parameters, a feasible solution can be found for conditions (13)–(15). The solution is listed as follows:(47)P1=6.78040.9268-0.0993-1.21560.92686.5528-1.2156-1.0043-0.0993-1.21566.78040.9268-1.2156-1.00430.92686.5528,P2=9.94910.82851.42490.20720.828510.31090.9803-0.29671.42490.98038.2046-1.12630.2072-0.2967-1.12637.4406,Q1=diag{12.7657,8.8809,12.7657,8.8809},Q2=diag{13.1103,22.4816,14.3887,12.1669}.According to Theorem 6, the controlled NDS can be stabilized by the partial and event-based couplings. To show this fact, the trajectories of the controlled NDS are depicted in Figure 3. It can be found that all the trajectories tend to zero.Figure 3
When the NDS is equipped with controllers, the trajectories of NDS tend to zero.Figure4 shows the sampling instants of the 2 channels of all the 4 agents of the NDS. The total number of the sampling instants of the NDS is 415. If the NDS employed the time-based sampling scheme with sampling period h=0.1, the sampling number would be 3200. Thus, the sampling number of the event-based sampling scheme is only 12.97% of that of the time-based sampling scheme. In other words, the event-based sampling scheme can save 87% communication resource.Figure 4
The sampling instants of all the channels of the NDS.


## 5. Conclusion

The stabilization problem of NDSs with partial and event-based couplings has been investigated. The communication among the agents has been assumed to suffer from two communication constraints: partial information transmission and event-based sampling. The former constraint leads to the information packet of each agent lack of integrity. However, the constraint of sampling makes the real-time information available at some discrete sampling instants. Furthermore, the sampling instants cannot be figured out in advance. Thus, the two constraints make the stabilization problem much harder. Furthermore, the control signals also suffered from these two constraints. By building the channel Laplacian matrices for different levels of information, the stabilization condition has been derived for the NDS with partial and event-based couplings. The numerical simulation has been given to show the effectiveness of our theoretical results. In this paper, the partial information transmission is considered as the communication constraint. However, some other constraints are also significant, such as the quantization and saturation. When we consider the event-based sampling scheme simultaneously, how would they impact the dynamical behaviour of NDSs? It is an interesting yet challenging problem. We will investigate it in our future work.



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*Source: 342465-2015-09-16.xml*