A Class of Distributed Event-Triggered Average Consensus Algorithms for Multi-Agent Systems
aa r X i v : . [ c s . M A ] N ov ARTICLE TEMPLATE
A Class of Distributed Event-Triggered Average ConsensusAlgorithms for Multi-Agent Systems
Ping Xu, Cameron Nowzari, Zhi Tian
Department of Electrical and Computer Engineering, George Mason University, Fairfax, VA,22030, USA
ARTICLE HISTORY
Compiled November 26, 2019
ABSTRACT
This paper proposes a class of distributed event-triggered algorithms that solve theaverage consensus problem in multi-agent systems. By designing events such that aspecifically chosen Lyapunov function is monotonically decreasing, event-triggeredalgorithms succeed in reducing communications among agents while still ensuringthat the entire system converges to the desired state. However, depending on thechosen Lyapunov function the transient behaviors can be very different. Moreover,performance requirements also vary from application to application. Consequently,we are instead interested in considering a class of Lyapunov functions such thateach Lyapunov function produces a different event-triggered coordination algorithmto solve the multi-agent average consensus problem. The proposed class of algorithmsall guarantee exponential convergence of the resulting system and exclusion of Zenobehaviors. This allows us to easily implement different algorithms that all guaranteecorrectness to meet varying performance needs. We show that our findings can beapplied to the practical clock synchronization problem in wireless sensor networks(WSNs) and further corroborate their effectiveness with simulation results.
KEYWORDS
Event-triggered control, distributed coordination, multi-agent consensus, varyingperformance needs, clock synchronization.
1. Introduction
The consensus problem of multi-agent systems where a group of agents arerequired to agree upon certain quantities of interest finds broad applica-tions in areas such as unmanned vehicles, mobile robots, and wireless sen-sor networks (WSNs) (Liang, Wang, Shen, & Liu, 2012; Olfati-Saber & Jalalkamali,2012; Peng, Wen, Rahmani, & Yu, 2015). Toward this problem, one effectiveand efficient method is the distributed event-triggered coordination approach,which was first proposed in (Dimarogonas & Johansson, 2009), and havebeen studied extensively over the last decades (Liu, Zhang, Yu, & Sun, 2018;Nowzari & Cort´es, 2016; Xie, Xu, Li, & Zou, 2015; Yi, Lu, & Chen, 2016), see refer-ences in (Ding, Han, Ge, & Zhang, 2017; Nowzari, Garcia, & Cort´es, 2019) for recentadvances and more details.The main idea behind distributed event-triggered algorithms is that the itera-tive communication between agents and their one-hop neighbors only happens whencertain conditions/events are triggered. Through skipping unnecessary communica-ions, the communication efficiency is increased, and at the same time the desiredproperties of the system are maintained. The triggering conditions of the event-triggered algorithms can be time-dependent (Seyboth, Dimarogonas, & Johansson,2013), state-dependent (Liu et al., 2018; Nowzari & Cort´es, 2014, 2016), ora combination of both (Girard, 2015; Sun, Huang, Anderson, & Duan, 2016;Yi, Liu, Dimarogonas, & Johansson, 2017). In general, the time-dependent thresh-olds are easy to design to exclude deadlocks (or Zeno behavior, mean-ing an infinite number of events triggered in a finite number of time pe-riod (Johansson, Egerstedt, Lygeros, & Sastry, 1999)), but require global informationto guarantee convergence to exactly a consensus state. While state-dependent thresh-olds are easier to design, these triggers might be risky to implement as Zeno behavior isharder to exclude. As the occurrence of Zeno behavior is impossible in a given physicalimplementation, the exclusion of it is therefore necessary and essential to guaranteethe correctness of an event-triggered algorithm.In this paper, we focus on developing event-triggered algorithms with state-dependent triggering thresholds that exclude the Zeno behavior. To be specific, anevent-triggered controller with state-dependent triggering thresholds can generally bedeveloped from a given Lyapunov function to maintain stability of a certain systemwhile reducing sampling or communication, using the given Lyapunov function as acertificate of correctness. In other words, all events are triggered based on how we wantthe given Lyapunov function to evolve in time. However, there are no formal guaran-tees on the gained efficiency. Moreover, it is known that a Lyapunov function is notunique for a given system, and each individual function may result in a totally differ-ent, but equally valid/correct triggering law. Consequently, there are many works thatpropose one such algorithm based on one function that all have the same guarantee:asymptotic convergence to a consensus state. That means there is no established wayto compare the performance of two different event-triggered algorithms that solve thesame problem. In particular, given two different event-triggered algorithms that bothguarantee convergence, their trajectories and communication schedules may be wildlydifferent before ultimately converging to the desired set of states. There are somenew works that are addressing exactly this topic (Borgers, Geiselhart, & Heemels,2017; Heijmans, Borgers, & Heemels, 2017; Khashooei, Antunes, & Heemels, 2017;Ramesh, Sandberg, & Johansson, 2016), which set the basis for this paper. Morespecifically, once established methods of comparing the performance of event-triggeredalgorithms against one another are developed, current available algorithms will likelybe revisited to optimize different types of performance metrics. In particular, we noticethat different algorithms are better than others in different scenarios when consideringmetrics such as convergence speed or total energy consumption. Therefore, instead oftrying to design only one event-triggered algorithm that simply guarantees conver-gence, we design an entire class of event-triggered algorithms that can be easily tunedto meet varying performance needs.Our work is motivated by (Nowzari & Cort´es, 2016) that solves the exact problemwe consider, i.e., design a distributed event-triggered algorithm with state-dependenttriggers for multi-agent systems over weight-balanced directed graphs. We first de-velop a distributed event-triggered algorithm based on an alternative Lyapunov can-didate function, which we name it as
Algorithm 2 . For the algorithm proposedby Nowzari and Cort´es (2016), we name it as
Algorithm 1 . Observing that thetwo algorithms result in different performance for different network topologies, wethen parameterize an entire class of Lyapunov functions from the two algorithmsand show how each individual function can be used to develop a
Combined Algo- ithm . More specifically, choosing any parameter λ ∈ [0 ,
1] yields an event-triggeredalgorithm that guarantees convergence. Changing λ can then help achieve varyingperformance goals while always guaranteeing stability. With the asymptotic conver-gence and exclusion of Zeno behavior for both Algorithm 1 and
Algorithm 2 ,we establish that the entire class of
Combined Algorithms also exclude Zenobehavior and guarantee convergence of the system. In addition to the theoreticanalysis, we also study the practical clock synchronization problem that exists inWSNs (Dimarogonas & Johansson, 2009), which is crucial especially when opera-tions such as data fusion, power management and transmission scheduling are per-formed (Kadowaki & Ishii, 2015; Wu, Chaudhari, & Serpedin, 2011). We use varioussimulations to illustrate the correctness and performance of our proposed algorithms.The rest of this paper is organized as follows. Section 2 introduces the preliminariesand Section 3 formulates the problem of interest. Section 4 first summarizes the relatedwork (Nowzari & Cort´es, 2016) and then proposes a novel strategy based on an alter-native Lyapunov function. Section 5 analyzes the non-Zeno behavior and convergenceproperty of the proposed strategy. The combined algorithms that are developed basedon the combined Lyapunov functions are proposed in Section 6, followed by a casestudy of clock synchronization in Section 7. Section 8 presents the simulation resultsand Section 9 concludes this work.
Notations: R , R > , R ≥ denote the set of real, positive real, and nonnegative realnumbers, respectively. N ∈ R N and N ∈ R N denote the N × k · k denotes the Euclidean norm forvectors or induced 2-norm for matrices. For a finite set S , | S | denotes its cardinality.
2. Preliminaries
Let G = {V , E , W } denote a weighted directed graph (or weighted digraph) that iscomprised of a set of vertices V = { , . . . , N } , directed edges E ⊂ V × V , and weightedadjacency matrix W ∈ R N × N ≥ . Given an edge ( i, j ) ∈ E , we refer to j as an out-neighbor of i and i as an in-neighbor of j . The sets of out- and in-neighbors of a givenagent i are N outi and N ini , respectively. The weighted adjacency matrix W satisfies w ij > i, j ) ∈ E and w ij = 0 otherwise. A path from vertex i to j is an orderedsequence of vertices such that each intermediate pair of vertices is an edge. A digraph G is strongly connected if there exists a path from all i ∈ V to all j ∈ V . The out- andin-degree matrices D out and D in are diagonal matrices whose diagonal elements are d outi = P j ∈N outi w ij , d ini = P j ∈N ini w ji , respectively. A digraph is weight-balanced if D out = D in , and the weighted Laplacianmatrix is given by L = D out − W .For a strongly connected and weight-balanced digraph, zero is a simple eigenvalueof L . In this case, we order its eigenvalues as λ = 0 < λ ≤ · · · ≤ λ N . Note thefollowing property will be of use later: λ ( L ) x T L T x ≤ x T L T Lx ≤ λ N ( L ) x T L T x. (1)Another property we need is the Young’s inequality (Hardy, Littlewood, & P´olya,3952), which states that given x, y ∈ R , for any ε ∈ R > , xy ≤ x ε + εy . (2)
3. Problem Statement
Consider the average consensus problem for an N -agent network described by a weight-balanced and strongly connected digraph G = {V , E , W } . Without loss of generality,we say that an agent i is able to receive information from neighbors in N outi andsend information to neighbors in N ini . Assume that all inter-agent communicationsare instantaneous and of infinite precision. Let x i denote the state of agent i ∈ V andconsider the single-integrator dynamics˙ x i ( t ) = u i ( t ) . (3)The well-known distributed continuous control law u i ( t ) = − P j ∈N outi w ij ( x i ( t ) − x j ( t )) (4)drives the states of all agents in the system to asymptotically converge to the aver-age of their initial states (Olfati-Saber & Murray, 2004). However, its implementationrequires all agents to continuously access their neighbors’ state information and keepupdating their own control signals, which is practically unrealistic in terms of bothcommunication and control. To relax both of these requirements, we adopt the modifieddistributed event-triggered control law (Dimarogonas, Frazzoli, & Johansson, 2012) u i ( t ) = − P j ∈N outi w ij (ˆ x i ( t ) − ˆ x j ( t )) , (5)where ˆ x i ( t ) to denote the last broadcast state of agent i and it remains constantbetween two broadcasts. That is, if we let t last be the last time at which agent i broadcasts its state information and t next be the next time it is going to broadcast,then ˆ x i ( t ) = x i ( t last ) for t ∈ [ t last , t next ). With this framework, neighbors of a givenagent are able to receive state information from it only when this agent decides tobroadcast its state information to them. After receiving the information from theirneighbors, agents then update their own control signals.Along with the above controller (5), each agent i is equipped with a triggeringfunction f i ( · ) that takes values in R . Our first objective is to identify triggers thatdepend on local information only, i.e., on the true state x i ( t ), its last broadcast stateˆ x i ( t ), and its neighbors last broadcast state ˆ x j ( t ) for j ∈ N i . Specifically, we need todesign triggering functions for each agent i ∈ V such that an event is triggered as soonas the triggering condition f i ( t, x i ( t ) , ˆ x i ( t ) , ˆ x j ( t )) > i to broadcast its state so that itsneighbors can update their states. To do so, the general steps are to identify a Lyapunovfunction for the system, and then derive triggering rules from the Lyapunov functionwhile maintaining the stability of the system and ensures asymptotic convergence toa consensus state. 4otice that a Lyapunov function is not unique for a given system, and each indi-vidual function may result in a totally different, but equally valid/correct triggeringlaw. Moreover, when considering metrics such as convergence speed or total energyconsumption, different algorithms are better than others in different scenarios. Sincethere is no established way to compare the performance of two different event-triggeredalgorithms that solve the same problem and performance requirements may vary fromapplication to application, therefore, our second objective is to design an entire classof event-triggered algorithms that can be easily tuned to meet varying performanceneeds. Before presenting our work, we first introduce the algorithm that motivates ourwork (Nowzari & Cort´es, 2016).
4. Distributed Trigger Design
The exact same problem of distributed event-triggered coordination for multi-agentsystems over weight-balanced digraphs has been studied by Nowzari and Cort´es(2016). As their findings are essential in developing our algorithms, we first summarizetheir algorithm and name it
Algorithm 1 .The event-triggered law proposed in (Nowzari & Cort´es, 2016) is Lyapunov-based,with the Lyapunov candidate function be V ( x ( t )) = ( x ( t ) − ¯ x ) T ( x ( t ) − ¯ x ) , (7)where x ( t ) = ( x ( t ) , ..., x N ( t )) T ∈ R N is the column vector of all agents’ states and¯ x = N P Ni =1 x i (0) N is the average of all initial conditions.The derivative of V ( x ( t )) takes the form˙ V ( x ( t )) = x T ( t ) ˙ x ( t ) − ¯ x T ˙ x ( t ) = − x T ( t ) L ˆ x ( t ) + ¯ x T L ˆ x ( t ) = − x T ( t ) L ˆ x ( t ) , (8)where ˙ x ( t ) = u ( t ) = − L ˆ x ( t ) is the compact vector-matrix form of equation (3) and(5), with ˆ x ( t ) = (ˆ x ( t ) , ..., ˆ x N ( t )) T ∈ R N the vector of last broadcast states of allagents. The second term ¯ x T L ˆ x ( t ) = 0 comes from the fact that the digraph G isweight-balanced, meaning TN L = T , therefore ¯ x T L ˆ x ( t ) = N P Ni =1 x i (0) TN L ˆ x ( t ) = 0.Expand (8) and apply Young’s inequality (2), ˙ V ( x ( t )) is upper bounded by˙ V ( x ( t )) ≤ − P Ni =1 P j ∈N outi w ij h (1 − a i )(ˆ x i ( t ) − ˆ x j ( t )) − e i ( t ) a i i , (9)where a i ∈ (0 ,
1) and e i ( t ) = ˆ x i ( t ) − x i ( t ) is the difference between agent i ’s lastbroadcast state and its current state at time t .To make sure that the Lyapunov function V ( x ( t )) is monotonically decreasing re-quires P j ∈N outi w ij h (1 − a i )(ˆ x i ( t ) − ˆ x j ( t )) − e i ( t ) a i i ≥ , for all agents i ∈ V at all times, which can be accomplished by enforcing e i ( t ) ≤ a i (1 − a i ) d outi P j ∈N outi w ij (ˆ x i ( t ) − ˆ x j ( t )) . (10)5t is found in (Nowzari & Cort´es, 2016) that by setting a i = 0 . f i ( e i ( t )) = e i ( t ) − σ i d outi P j ∈N outi w ij (ˆ x i ( t ) − ˆ x j ( t )) , (11)where σ i ∈ (0 ,
1) is a design parameter that affects the flexibility of the triggers.According to the triggering function (11), an event is triggered when f i ( e i ( t )) > f i ( e i ( t )) = 0 and φ i = P j ∈N outi w ij (ˆ x i ( t ) − ˆ x j ( t )) = 0.Basically, the trigger above makes sure that ˙ V ( x ( t )) is always negative as long as thesystem has not converged, therefore, Algorithm 1 guarantees all agents to converge tothe average of their initial states, i.e., lim t →∞ x ( t ) = ¯ x = N P Ni =1 x i (0) N , interestedreaders are referred to (Nowzari & Cort´es, 2016, Theorem 5.3) for more details. As we know, the Lyapunov function is not unique for the stability studying of thesame system, and each individual function may result a totally different triggeringlaw. Therefore, we propose a novel triggering strategy named
Algorithm 2 based onan alternative Lyapunov candidate function V ( x ( t )) = x ( t ) T L T x ( t ) . (12)The following result characterizes a local condition for all agents in the network suchthat the Lyapunov candidate function V ( x ( t )) is monotonically nonincreasing. Lemma 4.1.
For i ∈ V , with b i , c j < d outi ∀ i, j ∈ V , define e i ( t ) = ˆ x i ( t ) − x i ( t ) as inSection 4.1, with u i ( t ) given in (5) , then ˙ V ( x ( t )) ≤ − P Ni =1 h δ i u i ( t ) − (cid:16) d outi b i + d outi c i (cid:17) e i ( t ) (cid:17) , (13) where δ i , − d outi b i − P j ∈N outi w ij c j . (14) Proof.
See Appendix A.From Lemma 4.1, a sufficient condition to guarantee the proposed Lyapunov can-didate function V ( x ( t )) is monotonically decreasing is to ensure that δ i u i ( t ) − (cid:16) d outi b i + d outi c i (cid:17) e i ( t ) ≥ i ∈ V at all times, or e i ( t ) ≤ δ i b i c i ( b i + c i ) d outi (cid:16) P j ∈N outi w ij (ˆ x i ( t ) − ˆ x j ( t )) (cid:17) . (15)6he triggering function developed from Algorithm 2 is therefore derived as f i ( e i ( t )) = e i ( t ) − σ i δ i b i c i ( b i + c i ) d outi (cid:16) P j ∈N outi w ij (ˆ x i ( t ) − ˆ x j ( t )) (cid:17) , (16)where σ i ∈ (0 ,
1) is a design parameter that affects how flexible the trigger is andcontrols the trade-off between communication and performance. Setting σ i close to0 is generally greedy, meaning that the trigger is enabled more frequently and morecommunications are required, therefore makes agent i contribute more to the decreaseof the Lyapunov function V ( x ( t )), leading to a faster convergence of the network whilesetting the value of σ i close to 1 achieves the opposite results. Note that the rolesof b i , c i , c j are beyond system stabilization, they are also important to the trigger’sperformance. The larger value of δ i b i c i ( b i + c i ) d outi , the less communication shall be neededsince it means that the system is more error-tolerant. Corollary 4.2.
For agent i ∈ V with the triggering function defined in (16) , if thecondition f i ( e i ) ≤ is enforced at all times, then ˙ V ( x ( t )) ≤ − P Ni =1 (1 − σ i ) δ i ( P j ∈N outi w ij (ˆ x i ( t ) − ˆ x j ( t ))) . Similar as the work done in (Nowzari & Cort´es, 2016), to avoid the possibility thatagent i may miss any triggers, we define an event either by f i ( e i ( t )) > or (17) f i ( e i ( t )) = 0 and φ i = 0 (18)where φ i = ( P j ∈N outi w ij (ˆ x i ( t ) − ˆ x j ( t ))) .We also prescribe the following additional trigger as in (Nowzari & Cort´es, 2016) toaddress the non-Zeno behavior. Let t ilast be the last time at which agent i broadcastsits information to its neighbors. If at some time t ≥ t ilast , agent i receives informationfrom a neighbor j ∈ N outi , then agent i immediately broadcasts its state if t ∈ ( t ilast , t ilast + ε i ) , (19)where ε i < q σ i δ i b i c i ( b i + c i ) d outi (20)is a parameter selected to ensure the exclusion of Zeno behavior, and we will demon-strate how it is designed in the following section.We summarize the differences between Algorithm 1 proposedin (Nowzari & Cort´es, 2016) and
Algorithm 2 proposed here in Table 1. Once thetriggering function and parameters ε i are chosen for each agent, either algorithm canbe implemented using the coordination algorithm provided in Table 2.Note that both algorithms guarantee exponential convergence and the exclusion ofZeno behavior, as analyzed in Section 5 and in (Nowzari & Cort´es, 2016, Section 5).However, except for these similarities, we have no idea which algorithm works betterfor under varying performance need and initial conditions, which motivates our workin Section 6. 7 able 1. Difference between
Algorithm 1 and
Algorithm 2 .Triggering function Parameter design
Algorithm 1 f i ( e i ) , e i ( t ) − σ i d outi P j ∈N outi w ij (ˆ x i ( t ) − ˆ x j ( t )) ε i < q σ i d outi w max i |N outi | Algorithm 2 f i ( e i ) , e i − σ i δ i b i c i ( b i + c i ) d outi (cid:16) P j ∈N outi w ij (ˆ x i − ˆ x j (cid:17) ε i < r σ i δ i b i c i ( b i + c i ) d outi Table 2.
Distributed Event-Triggered Coordination Algorithm.
At all times t , agent i ∈ { , . . . , N } performs: if f i ( e i ( t )) > f i ( e i ( t )) = 0 and φ i = 0) then broadcast state information x i ( t ) and update control signal u i ( t ) end if if new information x j ( t ) is received from some neighbor(s) j ∈ N outi then if agent i has broadcast its state at any time t ′ ∈ [ t − ε i , t ) then broadcast state information x i ( t ) end if update control signal u i ( t ) end if5. Stability Analysis of Algorithm 2 In this section, we show that
Algorithm 2 guarantees that no Zeno behavior existsin the network executions. In addition, we show that when executing
Algorithm 2 ,all agents converge exponentially to the average of their initial states.
Proposition 5.1. (Non-Zeno Behavior) Consider the system (3) executing controllaw (5) . The triggering function is given by (16) . If the underlying digraph of thesystem is weight-balanced and strongly connected, then when executing the algorithmdescribed in Table 2, the system with any initial conditions will not exhibit Zeno be-havior.
Proof.
To prove that the system does not exhibit Zeno behavior, we need to showthat no agent broadcasts its state an infinite number of times in any finite time period.We divide the proof into two steps, the first step shows the existence of that finitetime period and gives its value; while in the second step, we show that no informationcan be transmitted an infinite number of times in that finite time period.
Step 1 : This step shows that if an agent does not receive new information from itsout-neighbors, its inter-events time is bounded by a positive constant.Assume that agent i ∈ V has just broadcast its state at time t , then e i ( t ) = 0. For t > t , while no new information is received, ˆ x i ( t ) and ˆ x j ( t ) remain unchanged. Giventhat ˙ e i = − ˙ x i , the evolution of the error is simply e i ( t ) = − ( t − t )ˆ z i , (21)where ˆ z i = P j ∈N outi w ij (ˆ x j − ˆ x i ). Since we are considering the case that no neighborsof agent i broadcast their states, therefore trigger (19) is irrelevant. We then needto find out the next time point t ∗ when f i ( e i ( t ∗ )) = 0 and agent i is triggered tobroadcast. This can be done following trigger (18). If ˆ z i = 0, no broadcasts will everhappen because e i ( t ) = 0 for all t ≥ t . Consider the case when ˆ z i = 0, using (21),8rigger (18) prescribes a broadcast at time t ∗ ≥ t that satisfies( t ∗ − t ) ˆ z i − σ i δ i b i c i ( b i + c i ) d outi ˆ z i = 0 , or equivalently ( t ∗ − t ) = σ i δ i b i c i ( b i + c i ) d outi . Therefore, we can lower bound the inter-events time by τ i = t ∗ − t = q σ i δ i b i c i ( b i + c i ) d outi , which explains our choice in (20). By this step, if none of agent i ’s neighbors broadcast,agent i will not be triggered infinitely fast. Next, we show that messages can not be sentinfinitely over a finite time period when one or more neighbors of agent i trigger(s). Step 2 : Same as
Step 1 , assume agent i has just broadcast its state at time t ,thus e i ( t ) = 0. Our reasoning is as follows:1) If no information is received by time t + ε i < t + τ i , then no trigger happensfor agent i .2) Let us then consider the situation that at least one neighbor of agent i broadcastsits information at some time t ∈ ( t , t + ε i ), which means that agent i would alsore-broadcast its information at time t due to trigger (19). Define I as the set in whichall agents have broadcast information at time t , then as long as no agent k ∈ I sendsnew information to any agent in I , agents in I will not broadcast new information forat least min j ∈ I τ j seconds, which includes the original agent i . As no new informationis received by any agent in I by time t + min j ∈ I ε j , there is no problem.3) Again consider the case that at least one agent k sends new information to someagent j ∈ I at time t ∈ ( t , t + min j ∈ I ε j ), then by trigger (19), all agents in I would also broadcast their state information at time t and agent k will now be addedto I . The remaining reasoning is just to repeat what has been reasoned, thus, theonly situation for infinite communications to occur in a finite time period is to have anetwork of infinite agents, which is impossible for the N -agent network we consider.Therefore, Step 1 and
Step 2 conclude that
Algorithm 2 excludes Zeno behaviorfor the network.Next we establish the global exponential convergence.
Theorem 5.2. (Exponential Convergence to Average Consensus). Given the sys-tem (3) executing Table 2 over a weight-balanced, strongly connected digraph, all agentsexponentially converge to the average of their initial states, i.e. lim t →∞ x ( t ) = ¯ x , where ¯ x = N P Ni =1 x i (0) N . Proof.
The triggering events (17) and (18) ensure that˙ V ( x ( t )) ≤ P Ni =1 ( σ i − δ i (cid:16) P j ∈N outi w ij (ˆ x i ( t ) − ˆ x j ( t )) (cid:17) . (22)To show that the convergence is exponential, we show that the evolution of V ( x ( t ))towards 0 is exponential. Omit the time stamp t for simplicity, and define σ max =9ax i ∈V σ i , δ max = max i ∈V δ i to further bound (22):˙ V ( x ) ≤ ( σ max − δ max N X i =1 (cid:16) X j ∈N outi w ij (ˆ x i − ˆ x j ) (cid:17) = ( σ max − δ max ˆ x T L T L ˆ x ≤ ( σ max − δ max λ ( L )ˆ x T L T ˆ x, where we use (1) to come up with the last inequality. Note that V ( x ) = 12 x T L T x = 12 (ˆ x − e ) T L T (ˆ x − e )= 12 (ˆ x T L T ˆ x − ˆ x T L T e − e T L T ˆ x + e T L T e ) ≤
12 (2ˆ x T L T ˆ x + 2 e T L T e ) ≤ ˆ x T L T ˆ x + k L kk e k . (23)Substitute (15) into (23), define d out min = min i ∈V d outi , b max = max i ∈V b i , c max =max i ∈V c i , b min = min i ∈V b i , and c min = min i ∈V c i , using (1), we haveˆ x T L T ˆ x + k L kk e k ≤ ˆ x T L T ˆ x + k L k σ max δ max b max c max ( b min + c min ) d out min ˆ x T L T L ˆ x ≤ ˆ x T L T ˆ x + k L k σ max δ max b max c max ( b min + c min ) d out min λ N ( L )ˆ x T L T ˆ x = (1 + 2 k L k σ max δ max b max c max λ N ( L )( b min + c min ) d out min )ˆ x T L T ˆ x. (24)Relate (23) with (24) gives˙ V ( x ) ≤ ( σ max − δ max λ ( L )ˆ x T L T ˆ x ≤ ( σ max − δ max λ ( L )2(1 + k L k σ max δ max b max c max λ N ( L )( b min + c min ) d out min ) x T L T x = ( σ max − b min + c min ) δ max λ ( L ) d out min ( b min + c min ) d out min + 2 k L k σ max δ max b max c max λ N ( L ) V ( x ) . (25)Substitute A = ( σ max − b min + c min ) δ max λ ( L ) d out min ( b min + c min ) d out min +2 k L k σ max δ max b max c max λ N ( L ) into (25), we have ˙ V ( x ( t )) ≤ AV ( x ( t )), therefore we conclude that V ( x ( t )) ≤ V ( x (0)) exp( At ) and the networkconverges exponentially to the average of its initial state.With the theoretical foundation of Algorithm 2 , we are now ready to propose aclass of event-triggered algorithms that can be tuned to meet varying performanceneeds under different scenarios. 10 . A Class of Event-Triggered Algorithms
As stated in Section 1, for a given system, there are many works studying event-triggered control using Lyapunov functions to reach the goal of maintaining the sta-bility of the system, while increasing the efficiency of the system. However, thereis very little work currently available that mathematically quantifies these benefits.Recently, some works began establishing results along this line (Antunes & Heemels,2014; Khashooei et al., 2017; Ramesh et al., 2016), still this area is in its infancy.In particular, there are not yet established ways to compare the performance of anevent-triggered algorithm with another. Consequently, many different algorithms canbe proposed to ultimately solve the same problem, while each algorithm is slightly dif-ferent and produces different trajectories. Specifically in our case,
Algorithm 1 and
Algorithm 2 solve the exact same problem, and offer the exact same guarantees, i.e.,they both exclude Zeno behavior and ensure asymptotic convergence of the network.So, which algorithm should we use? Moreover, we have found that depending on theinitial conditions and network topology, each algorithm may out-perform the otherin terms of different evaluation metrics. In any case, once these performance metricsbecome better researched, there will likely be more standard ways to mathematicallycompare the two different algorithms. Therefore, for now, instead of designing onlyone event-triggered algorithm for the system that only works better in one situation,we aim to design an entire class of algorithms that can easily be tuned to meet varyingperformance needs.We do this by parameterizing a set of Lyapunov functions rather than studyingonly a specific one. To the best of our knowledge, this paper is then a first study ofhow to design an entire class of algorithms that use different Lyapunov functions toguarantee correctness, with the intention of being able to use the best one at all times.In this paper, we utilize only two Lyapunov functions, however, we can also use asmany Lyapunov functions as we want and combine them all to develop the entire classof algorithms.Specifically, given any λ ∈ [0 , V λ ( x ( t )) = λV ( x ( t )) + (1 − λ ) V ( x ( t )) . (26)Accordingly, the derivative of V λ ( x ( t )) takes the form˙ V λ ( x ( t )) = λ ˙ V ( x ( t )) + (1 − λ ) ˙ V ( x ( t )) . (27)Following the steps of deriving the triggering functions in Section 4, the triggeringfunction developed based on the combined Lyapunov function (26) is given by f i ( e i ( t )) = e i ( t ) − σ i h λ d outi X j ∈N outi w ij (cid:16) ˆ x i ( t ) − ˆ x j ( t ) (cid:17) +(1 − λ )2 δ i b i c i ( b i + c i ) d outi (cid:16) X j ∈N outi w ij (ˆ x i ( t ) − ˆ x j ( t )) (cid:17) i . (28)We refer to the algorithm developed from the combined Lyapunov function as the Combined Algorithm parameterized by λ , with λ ∈ [0 , λ = 0 recovers Algorithm 2 and λ = 1 recovers Algorithm 1 .11imilarly, for the
Combined Algorithm , we use the following events to avoidmissing any triggers: f i ( e i ( t )) > , (29) f i ( e i ( t )) = 0 and φ i = 0 , (30)where, with a slight abuse of notation, φ i = λ d outi P j ∈N outi w ij (ˆ x i ( t ) − ˆ x j ( t )) + (1 − λ )2 δ i b i c i ( b i + c i ) d outi ( P j ∈N outi w ij (ˆ x i ( t ) − ˆ x j ( t ))) .The parameter that bounds the inter-events time and excludes Zeno behavior is alsodesigned: ε i < q λσ i d outi w max i |N outi | + − λ ) σ i δ i b i c i ( b i + c i ) d outi . Then, with the triggering function (28) and ε i defined above, the Combined Al-gorithm can also be implemented using Table 2.
Corollary 6.1.
Both
Algorithm 1 and
Algorithm 2 ensure all agents to exponen-tially converge to the average of their initial states with the proof that their Lyapunovfunctions converge exponentially. Therefore, as a linear combination of V ( x ( t )) and V ( x ( t )) , V λ ( x ( t )) also converges exponentially, which means that a network executingthe Combined Algorithm shall converge exponentially to the average of its initialstates.
To illustrate the correctness and effectiveness of
Algorithm 2 and the
CombinedAlgorithm , we introduce the fundamental clock synchronization problem that existsin wireless sensor networks (WSNs) as a case study.
7. Case Study: Clock Synchronization
WSNs are broadly applied in areas such as disaster management, border protection,and security surveillance, to name a few, thanks to their low-cost and collaborative na-ture (Abbasi & Younis, 2007; Gungor, Lu, & Hancke, 2010). However, the underlyinglocal clocks of these sensors are often in disagreement due to the imperfections of clockoscillators. To guarantee consistency in the collected data, it is crucial to synchronizethese clocks with high precision. In addition, as the small micro-processors embeddedin each sensor node are usually resource-limited (Gungor et al., 2010), energy-efficientcommunication protocols for clock synchronization are therefore desired.Quite a lot approaches have been proposed to solve this problem, ranging from cen-tralized to distributed, time-triggered to event-triggered, see (Carli & Zampieri, 2014;Chen, Li, Huang, & Tang, 2015; Choi & Shen, 2010; Garcia, Mou, Cao, & Casbeer,2017; Kadowaki & Ishii, 2015; Mar´oti, Kusy, Simon, & L´edeczi, 2004;Simeone & Spagnolini, 2007; Solis, Borkar, & Kumar, 2006) and references therein.To solve this fundamental problem, we propose to apply our event-triggered algo-rithms, i.e.,
Algorithm 2 and the
Combined Algorithm in this practical case.One of the most related works is done by Chen et al. (2015), where an event-triggeredalgorithm with state-dependent triggers is proposed. However, the virtual clocks theysynchronize are formed in a discrete manner, which may encounter abrupt changes.12he ability of avoiding abrupt changes is essential in clock synchronization since timediscontinuity due to these changes can cause serious faults such as missing importantevents (Sundararaman, Buy, & Kshemkalyani, 2005). While another event-triggeredalgorithm proposed by Garcia et al. (2017) does synchronize continuous-time virtualclocks, however, their time-dependent trigger design requires global information.Motivated by these two works, we introduce our state-dependent event-triggeredalgorithms that synchronize continuous-time virtual clocks.
Consider an N -sensor WSN whose topology is described by a strongly-connectedweight-balanced underlying digraph G = {V , E , W } , with V , E , W defined as in Sec-tion 2. Without loss of generality, we say that a sensor i is able to receive informationfrom its neighbors in N outi and send information to neighbors in N ini . Each sensor inthe network is equipped with a microprocessor with an underlying local clock l i ( t ),which is a function of the absolute time t ∈ R ≥ . Ideally, the local clocks should beconfigured as l i ( t ) = t so that the notion of time is consistent throughout the system.In reality (Kadowaki & Ishii, 2015), however, they are in the form of l i ( t ) = γ i t + o i , i = 1 , . . . , N, (31)where the unknown constants γ i ∈ R > and o i ∈ R represent the clock drift and offsetof i -th clock, respectively.As the absolute time t is not available, the clock drift γ i and offset o i can not becomputed directly. To synchronize the system, here we mean to synchronize the virtualclocks T i ( t ) of all sensors defined by (Kadowaki & Ishii, 2015) T i ( t ) = α i ( l i ( t )) l i ( t ) , i = 1 , . . . , N, (32)where α i ( l i ( t )) is the controlled drift and is a function of node i ’s local time l i ( t ).The clock synchronization is said to be achieved iflim t →∞ | T i ( t ) − T j ( t ) | = 0 , ∀ i, j ∈ { , . . . , N } . (33)For simple implementation, in this paper we consider the particular case where onlyclock drift is present, i.e., the clock offset o i = 0 for i = 1 , . . . , N . We also assume γ i ∈ [1 − ǫ γ , ǫ γ ], where ǫ γ is known. The local clocks are then given by l i ( t ) = γ i t, i = 1 , . . . , N. (34)Substitute (34) into (32) gives the expressions of virtual clocks T i ( t ) = γ i α i ( l i ( t )) t, i = 1 , . . . , N. (35)Note that the virtual clocks are continuous by definition, therefore the abruptchanges on the clocks are avoided.The dynamics of α i ( l i ( t )) is specified by d α i ( l i ( t ))d l i ( t ) = − P j ∈N outi w ij ( ˆ α i ( l i ( t )) − γ j γ i ˆ α j ( l j ( t ))) , (36)13here ˆ α i ( l i ( t )), ˆ α j ( l j ( t )) represent the last broadcast state values of sensor i and j attheir local time l i and l j , respectively. Though γ i and γ j can not be computed directly,the value of γ j γ i can be obtained as follows (Garcia et al., 2017): record the local timeof node i and node j when node i receives information from node j at two time points,say t m and t n , then a j a i can be computed using γ j γ i = l j ( t m ) − l j ( t n ) l i ( t m ) − l i ( t n ) . Note we only needthe local clock time, not the exact values of t m and t n .Define e i ( l i ( t )) = ˆ α i ( l i ( t )) − α i ( l i ( t )) as sensor i ’s state error, where α i ( l i ( t )) is itscurrent controlled drift. An event for sensor i is triggered as soon as the triggeringfunction f i ( l i ( t ) , α i ( l i ( t )) , ˆ α i ( l i ( t )) , ˆ α j ( l j ( t ))) > i to broadcast its current state α i ( l i ( t )) to its neighbors so that they can update their states accordingly. Our objectiveis to apply Algorithm 2 and the
Combined Algorithm so as to design triggeringfunctions (37) for each sensor with its locally available information so that the virtualclocks are synchronized, i.e., (33) is satisfied.
The event-triggered algorithms for clock synchronization are developed based on Lya-punov functions. To begin, let us first rewrite (35) as T i ( t ) = γ i α i ( l i ( t )) t = y i ( t ) t, (38)where y i ( t ) = γ i α i ( l i ( t )) is called the modified drift. It is clear that once consensus isachieved on the variables y i ( t ), the clock synchronization will be realized regardless ofthe individual values of γ i and α i ( l i ( t )).We then adopt the Lyapunov candidate functions proposed in Section 4, withthe modified drifts as variables, i.e., V ( y ( t )) = ( y ( t ) − ¯ y ) T ( y ( t ) − ¯ y ), V ( y ( t )) = y ( t ) T L T y ( t ), and V λ ( y ( t )) = λV ( y ( t )) + (1 − λ ) V ( y ( t )). As the algorithm de-velopment with different Lyapunov functions are similar, we only use V ( y ( t )) = y ( t ) T L T y ( t ) as an example to illustrate the derivation process.The dynamics of the modified drift y i ( t ) is derived as follows:˙ y i ( t ) = d α i ( l i ( t ))d l i ( t ) · d l i ( t )d t = γ i (cid:16) − X j ∈N outi w ij ( ˆ α i ( l i ( t )) − γ j γ i ˆ α j ( l j ( t ))) (cid:17) = − γ i (cid:16) X j ∈N outi w ij ( γ i ˆ α i ( l i ( t )) − γ j ˆ α j ( l j ( t )) (cid:17) = − γ i X j ∈N outi w ij (cid:16) ˆ y i ( t ) − ˆ y j ( t ) (cid:17) . (39)We then specify the following Lemma to upper bound the derivatives of V ( y ( t )). Lemma 7.1.
In clock synchronization, for i ∈ V , let b i , c j > for all i, j ∈ V (thesame b i , c j as in Lemma 4.1, define ν i ( t ) = P j ∈N outi w ij (ˆ y i ( t ) − ˆ y j ( t )) , and e yi ( t ) =14 y i ( t ) − y i ( t ) , then the derivative of V ( y ( t )) = y ( t ) T L T y ( t ) is upper bounded by ˙ V ( y ( t )) ≤ − P Ni =1 γ i (cid:20) δ i (cid:16) P j ∈N outi w ij (ˆ y i ( t ) − ˆ y j ( t )) (cid:17) − (cid:16) d outi b i + d outi c i (cid:17) e yi ( t ) (cid:21) , (40) where δ i is what defined in (14) . The proof is similar to the proof for Lemma 4.1 and is omitted due to space limit.From Lemma 7.1, we can see that as long as ˙ V ( y ( t )) < k Ly ( t ) k 6 = 0 hold, y i ( t ) achieves consensus, meaning lim t →∞ | y i ( t ) − y j ( t ) | = 0. Recall that T i ( t ) = y i ( t ) t ,therefore, lim t →∞ | T i ( t ) − T j ( t ) | = 0, proving that the synchronization on virtual clockscan be achieved.A sufficient condition to ensure that V ( y ( t )) is monotonically decreasing is e yi ( t ) ≤ δ i ( d outi b i + d outi c i ) (cid:16) X j ∈N outi w ij (ˆ y i ( t ) − ˆ y j ( t )) (cid:17) = 2 b i c i δ i ( b i + c i ) d outi (cid:16) X j ∈N outi w ij ( γ i ˆ α i ( l i ( t )) − γ j ˆ α j ( l i ( t ))) (cid:17) = 2 γ i b i c i δ i ( b i + c i ) d outi (cid:16) X j ∈N outi w ij ( ˆ α i ( l i ( t )) − γ j γ i ˆ α j ( l j ( t ))) (cid:17) . (41)With e yi ( t ) = γ i e i ( l i ( t )), we define the triggering function developed from Algo-rithm 2 as f i ( e i ( l i ( t ))) = e i ( l i ( t )) − b i c i δ i ( b i + c i ) d outi (cid:16) X j ∈N outi w ij ( ˆ α i ( l i ( t )) − γ j γ i ˆ α j ( l j ( t ))) (cid:17) . (42)To ensure no triggers are missed by sensor i , we define an event either by f i ( e i ( l i ( t ))) > or (43) f i ( e i ( l i ( t ))) = 0 and X j ∈N outi w ij ( ˆ α i ( l i ( t )) − γ j γ i ˆ α j ( l j ( t ))) = 0 . (44)Similarly, an additional trigger is prescribed to address the non-Zeno behavior. Let l lasti be the last time at which sensor i broadcasts its information to its neighbors. Ifat some time l i ( t ) ≥ l lasti , sensor i receives information from a neighbor j ∈ N outi , thenit immediately broadcasts its state if l i ( t ) ∈ ( l lasti , l lasti + ε ′ i ) , (45)where ε ′ i < s σ i b i c i δ i ( b i + c i ) d outi (46)whose design is as given in Proposition 5.1.The following result presents Algorithm 2 in the clock synchronization application.15 heorem 7.2.
For an N -sensor network over a weight-balanced digraph, assume onlyclock drift exists, i.e., o i = 0 , ∀ i ∈ V . With the virtual clocks (32) , dynamics givenin (36) , the distributed event-triggered consensus algorithm (42) - (46) ( Algorithm 2 )achieves asymptotic synchronization for the virtual clocks, i.e., (33) is satisfied.
We haven shown that
Algorithm 2 can be applied to the practical clock synchro-nization problem. Next, we show that the
Combined Algorithm can also be appliedto solve the clock synchronization problem. To do so, we first derive the triggering lawfor the clock synchronization problem from
Algorithm 1 as f i ( e i ( l i ( t ))) = e i ( l i ( t )) − σ i d outi X j ∈N outi w ij ( ˆ α i ( l i ( t )) − γ j γ i ˆ α j ( l j ( t ))) , (47)with an inter-event period bounded by ε ′ i < q σ i d outi w max i |N outi | .Then, with the triggering rules (42) - (47) and the analysis in Section 6, designingthe triggering function for the clock synchronization problem from the CombinedAlgorithm is straightforward. That is, f i ( e i ( l i ( t ))) = e i ( l i ( t )) − λσ i d outi X j ∈N outi w ij ( ˆ α i ( l i ( t )) − γ j γ i ˆ α j ( l j ( t ))) − − λ ) σ i δ i b i c i ( b i + c i ) d outi (cid:16) X j ∈N outi w ij ( ˆ α i ( l i ) − γ j γ i ˆ α j ( l j )) (cid:17) , (48)with an inter-event period bounded by ε ′ i < q λσ i d outi w max i |N outi | + − λ ) σ i δ i b i c i ( b i + c i ) d outi . Theorem 7.3.
For an N -sensor network over a weight-balanced digraph, assume onlyclock drift exists, i.e., o i = 0 , ∀ i ∈ V . With the virtual clocks (32) , dynamics givenin (36) , the distributed event trigging rule defined in (48) , then the Combined Algo-rithm achieves asymptotic synchronization for the virtual clocks when the triggeringcondition f i ( e i ( l i ( t ))) > or f i ( e i ( l i ( t ))) = 0 with e i ( l i ( t )) > is met. The proof of the theorem and the stability analysis, non-Zeno behavior exclusionare as given in Section 5, therefore are omitted.
8. Simulation Results
In this section, we apply
Algorithm 1 and
Algorithm 2 to the event-triggeredclock synchronization problem, to show the effectiveness of both algorithms. We thendemonstrate the performance of the proposed algorithms through several simulationsand show how either
Algorithm 1 or Algorithm 2 could be argued to be ‘better’given different network topology, which has set the basis for our introduction of the
Combined Algorithm to easily go between the two.We first show that both
Algorithm 1 and
Algorithm 2 are able to synchronizethe virtual clocks in WSNs. We consider four different network topologies, with theircorresponding weighted adjacency matrices listed in Table 3.The clock offset is 0 for all nodes, and the unknown clock drifts are γ =[0 .
65 0 .
79 0 .
91 1 .
25 1 . T . The evolution of the local clocks with respect to the absolute16 able 3. Four different networks.
Network 1: Random network Network 2: Ring network W = / / / / / / / /
60 0 1 / / W = / / / / / / / / / / Network 3: Complete network Network 4: Star network W = / / / / / / / / / / / / / / / / / / / / W = / / / / / / / / l ( t ) t (a) Local clocks T ( t ) T ( t ) t (b) Virtual clocks Figure 1.
Plots of the simulation results of the clock synchronization on Network 1. (a) The local clocksare the same for both algorithms. (b) Virtual clocks with the implementation of event-triggered control. Both
Algorithm 1 (top) and
Algorithm 2 (bottom) are able to synchronize the virtual clocks. time t is shown in Figure 1a. We can see that without any control, the local clockswill diverge.Then, we implement Algorithm 1 and
Algorithm 2 with the control law (36),triggering functions (47) and (42) developed from
Algorithm 1 and
Algorithm 2 ,respectively, to achieve clock synchronization. The involved parameters are set to be σ i = 0 .
5, and b i = c i = 0 . /d outi for all i ∈ { , . . . , N } . Both algorithms are able tosynchronize the virtual clocks on all four networks and we take the result on Network1 as an example and show the virtual clock evolution in Figure 1b. However, exceptfor the synchronization, we have no idea which algorithm performs better on otherevaluation metrics, for example, the convergence speed and total energy consumption.Also, the performance evaluation result may differ for different network topologies.Therefore, in the following simulations, we show the difference of the two differentalgorithms on four network topologies with different evaluation metrics.We plot the triggering instances of all nodes in the network when implementingthe event-triggered algorithms in Figure 2a, 2c, 2e, 2g. We notice that in general, thenumber of events triggered when implementing
Algorithm 1 is less than that whenimplementing
Algorithm 2 . We also plot the evolution of Lyapunov functions, i.e., V ( y ( t )) = ( y ( t ) − ¯ y ) T ( y ( t ) − ¯ y ), V ( y ( t )) = y ( t ) T L T y ( t ), and V λ ( y ( t )) = λV ( y ( t )) +171 − λ ) V ( y ( t )) for all networks with σ i = 0 . Network 3 , the Lyapunov function of
Algorithm 2 in the other three networksconverges faster than that of
Algorithm 1 . It is also noted that when the numberof events triggered when implementing
Algorithm 2 is noticeably larger than thatin implementing
Algorithm 1 , the convergence speed of the Lyapunov function in
Algorithm 2 is also noticeably faster than that in
Algorithm 1 . This is reasonable,since the more events are triggered, the more information is communicated in thenetwork, and the faster the consensus will be reached.The above simulations corroborate our argument that depending on the chosenevaluation metric, either algorithm can be argued to be ‘better’ than the other. Tobetter quantize/visualize the performance difference of the two algorithms and demon-strate our motivations for proposing the
Combined Algorithm , we then executingall algorithms with respect to varying σ . We evaluate these algorithms with four per-formance metrics, 1) the total number of events triggered, denoted by N e , 2) thetime needed for each network to reach a 99% convergence of the Lyapunov function,denoted by T con , 3) the total communication energy required to achieve a 99% con-vergence, denoted by E , and 4) the square of the H -norm of the system, denoted by C (Dezfulian, Ghaedsharaf, & Motee, 2018). The total communication energy neededis calculated by multiplying the power in units of milliwatt (mW) with T con , where weadopt the following power calculation model in units of mW (Martins et al., 2008): P = N X i =1 X j ∈{ ,...,N } ,j = i η . P i → j + ζ k α i ( l i ( t )) − α j ( l j ( t )) k , where ζ > η > P i → j is the power of the signal transmitted from agent i to agent j in units of dBmW.Similar as (Nowzari & Cort´es, 2012), we set η , ζ and P i → j to be 1. The square of the H -norm, C is defined by C := Z ∞ t =0 N X i =1 ( y i ( t ) − ¯ y ) dt, where y i ( t ) is the modified drift of each local clock and ¯ y is the average of the modifieddrift of the system.The involved parameters are set to be b i = c i = 0 . /d outi for all i ∈ { , . . . , N } .The same control law (36) is applied. For Algorithm 1 and
Algorithm 2 thatachieve clock synchronization, their triggering functions are given by (47) and (42),respectively. For the
Combined Algorithm , its triggering function is given by (48),with λ = 0 .
5. For each σ , we run 10 simulations with random clock drift that satisfiesto γ i ∈ (0 . , .
3) and obtain the average of N e , T con , E , and C in each simulation.From the top figures in Figure 3a, 3c, 3e, and 3g, we can see that for different σ , the total number of events triggered in the system when executing Algorithm2 is larger than that when executing
Algorithm 1 . On the other hand, from thebottom figures in Figure 3a, 3c, 3e, and 3g, we can see that the time needed to reacha 99% convergence of the system is usually much less when executing
Algorithm 2 ,with the only exception for the complete network, where both algorithms have similar18
EventsEvents t (a) Network 1, triggering instances V ( t ) t (b) Network 1, Lyapunov function EventsEvents t (c) Network 2, triggering instances V ( t ) t (d) Network 2, Lyapunov function EventsEvents t (e) Network 3, triggering instances V ( t ) t (f) Network 3, Lyapunov function EventsEvents t (g) Network 4, triggering instances V ( t ) t (h) Network 4, Lyapunov function Figure 2.
Plots of the triggering instances and the evolution of Lyapunov candidate functions on four networkswhen implementing both Algorithms. For figure (a), (c), (e), (g),
Algorithm 1 is on the top and
Algorithm2 is on the bottom. Algorithm 1Algorithm 2Combined Algorithm0 0.2 0.4 0.6 0.8 14567 N e T con σ (a) Network 1 -5 E C σ (b) Network 1 Algorithm 1Algorithm 2Combined Algorithm0 0.2 0.4 0.6 0.8 11.61.822.22.4 N e T con σ (c) Network 2 -5 E C σ (d) Network 2 Algorithm 1Algorithm 2Combined Algorithm0 0.2 0.4 0.6 0.8 111.21.41.6 N e T con σ (e) Network 3 -5 E C σ (f) Network 3 Algorithm 1Algorithm 2Combined Algorithm0 0.2 0.4 0.6 0.8 1468 N e T con σ (g) Network 4 -5 E C σ (h) Network 4 Figure 3.
Plots of different evaluation metrics. For figure (a), (c), (e), (g), top: total events triggered, bottom:convergence time (bottom); for figure (b), (d), (f), (h), top: energy consumption, bottom: H -norm squared. H -norm squared evaluates the distance of each local modified driftwith the average modified drift of the system, whose value therefore also indicates theconvergence speed of the system to some extent, see the bottom figures in Figure 3b,3d, 3f, and 3h. Therefore, depending on different network topologies and dependingon what performance metrics are most important for the application at hand, it maybe desirable to implement different types of event-triggered algorithms. Note thatthe Combined Algorithm can easily be tuned to approach either
Algorithm 1 or Algorithm 2 or anything in between to meet varying system needs by setting valuesfor λ . This also motivates our future work of adapting λ online to further improveperformance.
9. Conclusion
This paper proposes a class of distributed event-triggered communication and controllaw for multi-agent systems whose underlying directed graphs are weight-balanced.The class of algorithms are developed from a class of Lyapunov functions, each ofwhich is a linear combination (parameterized by λ ∈ [0 , λ defines a new Lyapunov function coupled with a new event-triggeredcoordination algorithm which uses that particular function to guarantee correctnessand is able to exclude the possibility of Zeno behavior. We show that the proposedentire class of event-triggered algorithms can be tuned to meet varying performanceneeds by adjusting λ . We also apply the proposed distributed event-triggered algo-rithms to solve the practical clock synchronization problem in WSNs. For the futureresearch, we will focus on developing a unified evaluation metric (which is a function ofdifferent performance needs) that can be used to evaluate the performance of differentalgorithms. In that way, the class of distributed algorithms will be developed from atunable algorithm to an adaptive algorithm. Funding
This work was supported by the NSF under Grant
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Appendix A. Proof of Lemma 4.1
Proof.
Omit the time stamp t for simplicity. The derivative of V ( x ) takes the form˙ V ( x ) = x T L T ˙ x. (A1)23ubstitute the vector form x = ˆ x − e into (A1), and expand it with (3), we have˙ V ( x ) = ˆ x T L T ˙ x − e T L T ˙ x = N X i =1 ( X j ∈N outi w ij (ˆ x i − ˆ x j ) u i − X j ∈N outi w ij ( e i − e j ) u i )= N X i =1 ( − u i − X j ∈N outi w ij e i u i + X j ∈N outi w ij e j u i )= N X i =1 ( − u i − d outi e i u i + X j ∈N outi w ij e j u i ) . (A2)For b i , c j >
0, applyYoung’s inequality (2) to the cross terms at the right hand sideof (A2) gives − d outi e i u i ≤ d outi b i e i + d outi b i u i , X j ∈N outi w ij e j u i ≤ X j ∈N outi w ij c j e j + X j ∈N outi w ij c j u i . Since the digraph is weight-balanced, the following equality holds: N X i =1 X j ∈N outi w ij c j e j = N X i =1 X j ∈N ini w ji c i e i = N X i =1 d ini c i e i = N X i =1 d outi c i e i . Combine the above inequalities and equality, we obtain an upper bound for ˙ V ( x ):˙ V ( x ) ≤ N X i =1 (cid:16) − u i + d outi e i b i + d outi b i u i d outi e i c i + X j ∈N outi w ij c j u i (cid:17) = − N X i =1 "(cid:16) − d outi b i − X j ∈N outi w ij c j (cid:17) u i − (cid:16) d outi b i + d outi c i (cid:17) e i = − N X i =1 " δ i u i − (cid:16) d outi b i + d outi c i (cid:17) e i , (A3)with δ i defined in (14). To ensure δ i >
0, we require b i , c j < d outiouti