A Class of Distributed Event-Triggered Average Consensus Algorithms for Multi-Agent Systems

AR TICLE TEMPLA TE A Class of Distributed Even t-T riggered Av erage Consensus Algorithms for Multi-Agen t Systems Ping Xu, Cameron No wzari, Zhi Tian Department of E lectrical and Computer Engineer ing , George Maso n Universit y , F a ir fax, V A, 22030 , USA AR TICLE HISTOR Y Compiled Nov ember 27, 2024 ABSTRACT This paper proposes a class of distribut ed even t- triggered algorithms th at solve the a verage consensus problem in multi-agen t systems. By designing even ts such that a sp ecifically c hosen Lyapuno v function is monotonically decreasing, even t-triggered algorithms succeed in reducing communicatio ns among agents while still ensuring that the entire system con verges to th e desired state. How ever, dep ending on th e chos en Lyapunov function the transient b ehaviors can b e v ery different. Moreo ver, p erformance requirements also v ary from application to app lication. Consequently , w e are instead interested in considering a class of Lyapuno v functions such that eac h Ly apunov function produ ces a different even t- t riggered co ordination algorithm to solv e the m ulti-agent a verage consensus problem. The proposed class of algorithms all guaran t ee exp onential conv ergence of the resulting system and exclusion of Zeno b eha viors. This allo ws us to eas ily implemen t different algorithms that all guaran tee correctness to meet v arying performance needs. W e sho w that our fi ndings can b e applied to the practical clock sync hronization problem in wireless sensor n etw orks (WSNs) and furth er corrob orate their effective ness with sim ulation results. KEYWO RDS Even t- t riggered contro l, distributed co ordin ation, multi-agen t consensus, v arying p erformance n eeds, clo ck synchronization. 1. In tro duction The consensus problem of m ulti-agen t systems where a group of a gen ts are required to agree up on c ertain quan tities of in terest fin ds broad applica- tions in areas suc h as unmann ed v ehicles, mobile rob ots, and wireless sen- sor net works (WSNs) (Liang, W ang, Shen, & Liu, 2012; Olfati-Sab er & Jalalk amali, 2012; P eng, W en, Rahmani, & Y u, 2015). T o w ard this problem, one e ffectiv e and efficien t metho d is the distributed ev ent-t riggered coordin ation approac h, whic h w as fi rst p r op osed in (Dimarogonas & Johans s on, 20 09), and ha ve b een studied extensiv ely o ver the last decades (Liu, Zhang, Y u, & Su n, 2018; No wzari & Cort ´ es, 2016; Xie, Xu, Li, & Zou, 2015; Yi, Lu, & Chen, 2016), see refer- ences in (D ing, Han, Ge, & Z hang, 20 17; No w zari, Garcia, & Cort´ e s, 2019) for rece n t adv ances and more details. The main idea b ehin d distributed ev ent-t riggered algo rithms is t hat the itera- tiv e comm un ication b etw een agent s and their one-hop n eigh b ors only happ en s wh en certain cond itions/ev en ts are trig gered. Through skipping u nnecessary comm unica- tions, the comm unication efficie ncy is in creased, and a t the same time the desired prop erties of the system are main tained. The triggering conditions o f th e ev en t- triggered algorithms can b e time-depend en t (Seyb oth, Dimarog onas, & Joh an s son, 2013), sta te-dep endent (Liu et al., 2018; No wzari & C ort ´ es , 2014, 2016), or a com b ination of b oth (Gi rard, 2015; Sun, Huang, Anderson, & Duan, 2016; Yi, Liu, Dimarogonas, & Johanss on, 2017). In general, the time- dep end en t thresh - olds are e asy to desig n to excl ude deadlo c ks (or Zeno b eha vior, mea n- ing an infinite num b er of ev ent s triggered in a finite num b er of time p e- rio d (Johans s on, Egerstedt, Lygeros, & Sastry, 1999)), but require global information to guaran tee con v er gence to exactly a c onsensus state. While state-dep end en t thresh - olds are easier to design, these triggers migh t b e risky to implement as Zen o beh a vior is harder to exclude. As the o ccurrence of Zeno b eha vior is imp ossible in a giv en p h ysical implemen tation, th e exclusion of it is therefore necessary and essen tial to guarantee the correctness of an ev en t-triggered algorithm. In this pap er, w e fo cus on dev eloping ev en t-triggered algorithms with state- dep endent triggering thresholds that exclude the Zeno b eha vior. T o b e sp ecific, an ev ent-trig gered controlle r with state-dep endent triggering thresholds can ge nerally b e dev elop ed from a given L yapuno v function to mainta in stabilit y of a certain system while redu cing sampling or communicati on, using the giv en Lyapuno v function as a certificate of correctness. In other w ord s, all ev ents are triggered based on ho w we wan t the giv en Lyapuno v f unction to ev olv e in time. Ho w ev er, there are no formal guaran- tees on the gained efficiency . Moreo ver, it is kno w n that a Lya punov function is n ot unique for a giv en system, and eac h in dividual function ma y result in a tota lly d iffer- en t, but equally v alid/correct triggering law. Consequently , there are man y works that prop ose one such algorithm based on one function that all hav e the same guaran tee: asymptotic con verge nce to a consensus state. That means there is no established w a y to compare the p erformance of t wo differen t ev ent -triggered algorithms that solve the same problem. In particular, giv en t wo different even t-triggered algorithms that b oth guaran tee co n v ergence, their tra jectories and communicati on sc hedu les may b e wildly differen t b efore u ltimately conv erging to the desired set of states. There a re some new w orks that are addressing exactly this topic (Borgers, Geiselhart, & Heemels, 2017; He ijmans, Borgers, & Heemels , 2 017; Khasho o ei, An tun es, & Heemel s, 2017; Ramesh, Sandb erg, & Johansson, 2016), whic h set the basis for this pap er. More sp ecifically , once established metho d s of comparing the p erformance of ev ent-t riggered algorithms against one an other are dev elop ed, curr ent a v ailable algorithms will lik ely b e revisited to optimize different t yp es of p erformance metrics. In particular, w e n otice that d ifferen t algorithms are b etter than others in d ifferen t scenarios when considering metrics suc h as con v ergence s p eed o r total energy consumption. Therefore, instead of trying to design only one ev ent-t riggered algorithm that sim p ly guaran tees conv er- gence, we design a n en tire c lass of even t-triggered algorithms that can b e ea sily tu n ed to meet v arying p erform an ce needs. Our wo rk is motiv ated by (No wzari & Cort´ e s , 2016) that solve s the exact pr oblem w e consider, i.e., design a distr ib uted ev ent-trig gered algorithm with state-dep endent triggers for multi- agen t systems o v er weigh t-balanced directe d graph s. W e first de- v elop a distributed ev en t-triggered algorithm b ased on an alternativ e Lyapuno v can- didate fu nction, which w e name it as Algorithm 2 . F or the alg orithm prop osed b y No wzari and C ort´ es (2016), we name it a s Algorithm 1 . Observing that the t wo algo rithms result in d ifferen t p erformance for different net w ork top ologies, w e then parameterize an en tire class o f Ly apuno v fun ctions from the t w o a lgorithms and sh o w how eac h individ u al function can b e used to devel op a Com bined Algo- 2 rithm . More sp ecifically , c h o osing any parameter λ ∈ [0 , 1] yields an ev ent-trig gered algorithm that guarante es con verge nce. Ch anging λ can then help ac hiev e v arying p erformance goals wh ile alw a ys guaran teeing stabilit y . With the asymptotic con ver- gence and exclusion of Zeno b ehavio r for b oth Algorithm 1 and Algorithm 2 , w e establish that the entire class of Com bined Algorithms also exclude Zen o b eha vior and guaran tee con v er gence o f the system. In addition to the theoretic analysis, we also study t he pr actica l cloc k syn c hr onization pr oblem that exists in WSNs (Dimarogonas & Johansson, 2009), which is crucial esp ecially when op era- tions suc h as data fusion, p o w er management and trans mission scheduling are p er- formed (Kad ow aki & I shii, 2015; W u, C haudhari, & Serp edin, 2011). W e use v arious sim ulations to illustrate the correctness and p erformance of o ur pr op osed algorithms. The rest of this p ap er is organized as follo ws. Section 2 introdu ces the preliminaries and Section 3 form ulates the problem of interest. Section 4 fi rst summarizes th e relat ed w ork (No wzari & Cort´ e s, 2016) and then prop oses a no vel strategy based o n an alter- nativ e Lya punov fu nction. Section 5 analyzes the non-Zeno b eha vior and conv ergence prop erty of the pr op osed str ategy . The com b in ed algorithms th at are develo p ed based on the com b in ed Lya punov functions are prop osed in S ection 6, follo w ed b y a case study of clo ck syn c h r onization in Section 7. S ection 8 pr esents the simula tion results and Section 9 concludes this work. Notations: R , R > 0 , R ≥ 0 denote the set of real, p ositiv e rea l, and nonnegativ e real n umbers, resp ectiv ely . 1 N ∈ R N and 0 N ∈ R N denote the N × 1 column v ectors with en tries all equal to one and zero, resp ectiv ely . k · k den otes the Euclidean norm for v ectors or indu ced 2- norm for matrices. F or a fi nite set S , | S | denotes its cardinalit y . 2. Preliminaries Let G = {V , E , W } den ote a weig h ted directed graph (or weig h ted digraph) that is comprised of a set of v ertices V = { 1 , . . . , N } , d irected edges E ⊂ V × V , and weigh ted adjacency ma trix W ∈ R N × N ≥ 0 . Give n an edge ( i , j ) ∈ E , w e r efer to j as a n out- neigh b or o f i and i as an in-neighbor of j . Th e sets of out- and in-neighbors of a give n agen t i are N out i and N in i , resp ectiv ely . The weig h ted adjacency matrix W satisfies w ij > 0 if ( i, j ) ∈ E and w ij = 0 otherwise. A path from v ertex i to j is an ordered sequence of ve rtices suc h that e ac h inte rmediate pair of v ertices is an edge. A digraph G is strongly connected if there exists a path f rom all i ∈ V to a ll j ∈ V . Th e out- and in-degree matrices D out and D in are diagonal matrices whose diagonal elemen ts are d out i = P j ∈N out i w ij , d in i = P j ∈N in i w j i , resp ectiv ely . A digraph is w eight- balanced if D out = D in , and the w eigh ted Laplacian matrix is giv en by L = D out − W . F or a strongly connected and weigh t-balanced digraph, zero is a s imple eigen v alue of L . I n th is case, w e order its eigen v alues as λ 1 = 0 < λ 2 ≤ · · · ≤ λ N . Note the follo wing prop er ty will b e o f us e later: λ 2 ( L ) x T L T x ≤ x T L T Lx ≤ λ N ( L ) x T L T x. (1) Another prop erty w e n eed is th e Y oung’s inequality (Hardy , Littlewoo d, & P´ ol y a, 3 1952), whic h states that giv en x, y ∈ R , for any ε ∈ R > 0 , xy ≤ x 2 2 ε + εy 2 2 . (2) 3. Problem Statement Consider th e av erage consensus prob lem for an N -age nt n et work describ ed by a w eight- balanced an d strongly connected digraph G = {V , E , W } . Without loss of generalit y , w e sa y that an agen t i is able to receiv e information from n eighb ors in N out i and send information to neigh b ors in N in i . Assu m e that all inte r-agen t c omm unications are instan taneous and of infinite precision. Let x i denote the state of ag en t i ∈ V and consider the single-in tegrator dyn amics ˙ x i ( t ) = u i ( t ) . (3) The w ell-kno wn distribu ted con tin u ous control la w u i ( t ) = − P j ∈N out i w ij ( x i ( t ) − x j ( t )) (4) driv es the states of all agen ts in the system to asymptotically conv erge to the a ver- age of t heir initial sta tes (Olfati -Sab er & Murr a y, 20 04). How ev er, its implemen tation requires all agen ts to con tinuously access their neigh b ors’ state information and k eep up d ating their o w n con trol signals, whic h is p ractically unr ealistic in terms of b oth comm u nication and cont rol. T o r elax b oth of these requirements, we ad op t the mod ified distributed ev en t-triggered control la w (Dimarogonas, F razzoli, & Johansson, 2012) u i ( t ) = − P j ∈N out i w ij ( ˆ x i ( t ) − ˆ x j ( t )) , (5) where ˆ x i ( t ) to denote the last broadcast state of age nt i and it remains constan t b et w een t wo broadcasts. That is, if we let t last b e the last time at whic h agen t i broadcasts its state information and t next b e the next time it is going to b roadcast, then ˆ x i ( t ) = x i ( t last ) for t ∈ [ t last , t next ). With this framew ork, neighbors of a giv en agen t are able to r eceiv e state inform ation from it only wh en th is agen t decides to broadcast its state information to them. After r eceiving the inf ormation from their neigh b ors, agen ts then up date their o wn control signals. Along with the ab o v e con troller (5), eac h agen t i is equipp ed with a trigg ering function f i ( · ) th at tak es v alues in R . Our first ob jectiv e is to identify triggers that dep end on lo cal in formation only , i.e., on the true state x i ( t ), its last br oadcast state ˆ x i ( t ), and its n eigh b ors last br oadcast state ˆ x j ( t ) for j ∈ N i . Sp ecifically , w e need to design triggering fun ctions for eac h agen t i ∈ V such that an eve nt is triggered as soon as the triggering condition f i ( t, x i ( t ) , ˆ x i ( t ) , ˆ x j ( t )) > 0 (6) is fulfi lled. Th e tr iggered ev ent then drives agent i to broadcast its state so that its neigh b ors can up date their stat es. T o d o so, the general steps are to identify a Lyapuno v function for the system, and then deriv e triggering r ules from the Lyapuno v function while main taining the stabilit y of the system and ensures asymptotic con vergence to a consensus state. 4 Notice that a Lyapuno v fu n ction is n ot un ique for a giv en system, and eac h indi- vidual function ma y r esu lt in a totally different, but equally v alid/correct tr iggering la w. Moreo ve r, wh en considering metrics suc h as con v ergence sp eed or total energy consumption, different algorithms are b etter than others in differen t scenarios. S ince there is no established w a y to compare the p er f ormance of t w o differen t ev en t-triggered algorithms that solve the same problem and p erformance requ iremen ts ma y v ary from application to application, therefore, our second ob jectiv e is to d esign an en tire class of even t-triggered algorithms that can b e easily tuned to m eet v aryin g p erformance needs. Before present ing our work, we fi rst introd u ce the algorithm that motiv ates our w ork (No w zari & Cort ´ es, 2016). 4. Distributed T rigger Design 4.1. R elate d wor k The exact same problem of distr ib uted eve n t-triggered co ord ination for m ulti-agen t systems ov er we igh t-balanced d igraphs has b een studied by No w zari and Cort ´ es (2016). As their fi ndings are essen tial in dev eloping our algorithms, w e first summarize their algorithm and name it Algorithm 1 . The ev en t-triggered la w pr op osed in (No w zari & Cort´ e s , 2016) is Ly apuno v-based, with the Ly apunov c andidate f u nction b e V 1 ( x ( t )) = 1 2 ( x ( t ) − ¯ x ) T ( x ( t ) − ¯ x ) , (7) where x ( t ) = ( x 1 ( t ) , ..., x N ( t )) T ∈ R N is the column v ector of all agen ts’ states and ¯ x = 1 N P N i =1 x i (0) 1 N is the a verage of all initial conditions. The deriv ativ e of V 1 ( x ( t )) tak es the form ˙ V 1 ( 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 v ector-matrix form of equ ation (3) and (5), with ˆ x ( t ) = ( ˆ x 1 ( t ) , ..., ˆ x N ( t )) T ∈ R N the vect or of last broadcast states of all agen ts. Th e second term ¯ x T L ˆ x ( t ) = 0 come s from the f act that the digraph G is w eigh t-balanced, meaning 1 T N L = 0 T , therefore ¯ x T L ˆ x ( t ) = 1 N P N i =1 x i (0) 1 T N L ˆ x ( t ) = 0. Expand (8) and apply Y oung’s inequalit y (2), ˙ V 1 ( x ( t )) is upp er b ounded b y ˙ V 1 ( x ( t )) ≤ − 1 2 P N i =1 P j ∈N out i w ij h (1 − a i )( ˆ x i ( t ) − ˆ x j ( t )) 2 − e 2 i ( t ) a i i , (9) where a i ∈ (0 , 1) a nd e i ( t ) = ˆ x i ( t ) − x i ( t ) is the differen ce b et wee n agen t i ’s last broadcast state and its cur r en t state at time t . T o make su re that the L yapuno v fun ction V 1 ( x ( t )) is monotonically d ecreasing re- quires P j ∈N out i w ij h (1 − a i )( ˆ x i ( t ) − ˆ x j ( t )) 2 − e 2 i ( t ) a i i ≥ 0 , for all agen ts i ∈ V at all times, whic h can b e accomplished by e nforcing e 2 i ( t ) ≤ a i (1 − a i ) d out i P j ∈N out i w ij ( ˆ x i ( t ) − ˆ x j ( t )) 2 . (10) 5 It is found in (No wzari & Cort ´ es, 2016) that by setting a i = 0 . 5 for all agen ts, th e trigger design will b e optimal. Therefore, the triggering function in (No w zari & Cort´ es, 2016) is defined as f i ( e i ( t )) = e 2 i ( t ) − σ i 4 d out i P j ∈N out i w ij ( ˆ x i ( t ) − ˆ x j ( t )) 2 , (11) where σ i ∈ (0 , 1) is a design parameter t hat affects the flexibilit y of the t riggers. According to the triggering function (11), an eve nt is tr iggered wh en f i ( e i ( t )) > 0 or when f i ( e i ( t )) = 0 and φ i = P j ∈N out i w ij ( ˆ x i ( t ) − ˆ x j ( t )) 2 6 = 0. Basically , the trigg er ab o ve makes sure th at ˙ V 1 ( x ( t )) is alwa ys negativ e as long as the system h as n ot con v erged, therefore, Algorithm 1 guaran tees all agents to con verge to the av erage of their initial states, i.e., lim t →∞ x ( t ) = ¯ x = 1 N P N i =1 x i (0) 1 N , intereste d readers are referred to (No w zari & Cort´ es, 2016, Theorem 5.3) for more d etails. 4.2. Pr op ose d new algor ithm As w e kno w, the Lyapuno v function is not unique for the stabilit y stud ying of the same system, and eac h individual function ma y resu lt a tota lly differen t triggering la w. Ther efore, w e prop ose a no v el tr iggering strate gy named Algorithm 2 based on an alternativ e Ly apunov candidate function V 2 ( x ( t )) = 1 2 x ( t ) T L T x ( t ) . (12) The follo wing r esult c h aracterizes a lo cal condition f or all agen ts in the net work such that the Lya punov c andidate fun ction V 2 ( x ( t )) is monotonically nonincreasing. Lemma 4.1. F or i ∈ V , with b i , c j < 1 d out i ∀ i, j ∈ V , define e i ( t ) = ˆ x i ( t ) − x i ( t ) as in Se ction 4.1, with u i ( t ) given in (5) , then ˙ V 2 ( x ( t )) ≤ − P N i =1 h δ i u 2 i ( t ) −  d out i 2 b i + d out i 2 c i  e 2 i ( t )  , (13) wher e δ i , 1 − d out i b i 2 − P j ∈N out i w ij c j 2 . (14) Pr o of. See App endix A. F r om Lemma 4.1, a su fficien t condition to guaran tee the p r op osed Ly apu no v can- didate function V 2 ( x ( t )) is monotonically decreasing is to ensu r e that δ i u 2 i ( t ) −  d out i 2 b i + d out i 2 c i  e 2 i ( t ) ≥ 0 for all agen ts i ∈ V at all times, or e 2 i ( t ) ≤ 2 δ i b i c i ( b i + c i ) d out i  P j ∈N out i w ij ( ˆ x i ( t ) − ˆ x j ( t ))  2 . (15) 6 The triggering function devel op ed from Algorithm 2 is th er efore derived as f i ( e i ( t )) = e 2 i ( t ) − 2 σ i δ i b i c i ( b i + c i ) d out i  P j ∈N out i w ij ( ˆ x i ( t ) − ˆ x j ( t ))  2 , (16) where σ i ∈ (0 , 1) is a design parameter that affects how flexible the trigger is and con trols th e tr ade-off b et ween communicati on and p erform ance. S etting σ i close to 0 is generally greedy , meaning that the trigger is enabled more frequently and more comm u nications are required, therefore ma k es agent i co n tribute more to the d ecrease of the Ly apunov fun ction V 2 ( x ( t )), leading to a faster co n v ergence of the net w ork w hile setting the v alue of σ i close to 1 ac h iev es the opp osite results. Note that the roles of b i , c i , c j are b eyo nd sy s tem stabilizatio n, they are also imp ortan t to the trigger’s p erformance. T he larger v alue of 2 δ i b i c i ( b i + c i ) d out i , the less comm unication shall b e needed since it means that the system is more error-toleran t. Corollary 4.2. F or agent i ∈ V w ith the triggering function define d i n (16) , if the c ondition f i ( e i ) ≤ 0 is enfor c e d at al l times, then ˙ V 2 ( x ( t )) ≤ − P N i =1 (1 − σ i ) δ i ( P j ∈N out i w ij ( ˆ x i ( t ) − ˆ x j ( t ))) 2 . Similar as the w ork done in (No wzari & Cort ´ es , 20 16), to av oid the p ossibilit y that agen t i ma y miss any trigg ers, we define an ev en t either b y f i ( e i ( t )) > 0 or (17) f i ( e i ( t )) = 0 and φ i 6 = 0 (18) where φ i = ( P j ∈N out i w ij ( ˆ x i ( t ) − ˆ x j ( t ))) 2 . W e a lso prescrib e the follo wing add itional trigger as in (No wzari & Cort ´ es, 2016) to address the non-Z en o b eha vior. Let t i last b e the last time at w h ic h agen t i broadcasts its information to its neig hbors. If at some time t ≥ t i last , agen t i receiv es in formation from a neigh b or j ∈ N out i , then agen t i immediately broadcasts its state if t ∈ ( t i last , t i last + ε i ) , (19) where ε i < q 2 σ i δ i b i c i ( b i + c i ) d out i (20) is a parameter selected to ensure the exclusion of Zeno b ehavior, and w e will demon- strate how i t is designed in the follo w ing section. W e summarize the differences b et wee n Algorithm 1 prop osed in (No wzari & Cort´ es, 2016) and Algorithm 2 p r op osed h ere in T able 1. Once the triggering fun ction and parameters ε i are c h osen for eac h agen t, either algorithm can b e implemen ted usin g the co ordination algorithm pro vided in T able 2. Note that b oth algorithms guarante e exp onen tial con ve rgence and the exclusion of Zeno b ehavio r, as analyzed in Section 5 and in (Nowza ri & C ort ´ es , 2016, Section 5). Ho wev er, except for these s im ilarities, we ha v e no idea whic h algorithm w orks b etter for und er v arying p erform an ce n eed and initial conditions, whic h motiv ates our w ork in Section 6. 7 T able 1. Difference betw een Algori thm 1 and Algori thm 2 . T r iggering function Pa rameter design Algorithm 1 f i ( e i ) , e 2 i ( t ) − σ i 4 d out i P j ∈N out i w ij ( ˆ x i ( t ) − ˆ x j ( t )) 2 ε i < q σ i 4 d out i w max i |N out i | Algorithm 2 f i ( e i ) , e 2 i − 2 σ i δ i b i c i ( b i + c i ) d out i  P j ∈N out i w ij ( ˆ x i − ˆ x j  2 ε i < r 2 σ i δ i b i c i ( b i + c i ) d out i T able 2. Distributed Eve nt -T riggered Coor di nation Algorithm. A t all times t , agen t i ∈ { 1 , . . . , N } p erforms: 1: if f i ( e i ( t )) > 0 or ( f i ( e i ( t )) = 0 and φ i 6 = 0) then 2: broadcast state information x i ( t ) and up date control signal u i ( t ) 3: end if 4: if new information x j ( t ) is r eceiv ed from some neighbor(s) j ∈ N out i then 5: if agen t i has broadcast its state at an y time t ′ ∈ [ t − ε i , t ) then 6: broadcast state information x i ( t ) 7: end if 8: up d ate con trol signal u i ( t ) 9: end if 5. Stabilit y Analysis of Algorithm 2 In this section, we show that Algorithm 2 guaran tees that no Zeno b ehavio r exists in the net w ork executions. In addition, w e sho w that when executing Algorithm 2 , all agen ts con verge exp onential ly to th e a v erage of th eir initial states. Prop osition 5.1. (Non-Zeno Behavior) Co nsider the system (3) exe cuting c ontr ol law (5) . The triggering function is given by (16 ) . If t he underlying digr aph of the system is weig ht-b alanc e d and str ongly c onne cte d, then when exe cuting the algorithm describ e d in T able 2, the system with any initial c onditions wil l not exhibit Zeno b e- havior. Pr o of. T o p ro ve that the sys tem do es not exh ibit Zen o b eha vior, we n eed to show that no agen t broadcasts its state an infi nite n umber of times in any finite time p erio d. W e divide the p ro of in to t wo steps, the first step sho ws the existence of that finite time p erio d and give s its v alue; w h ile in the seco nd ste p, w e sho w that no in formation can b e transmitted an infi n ite n umb er of time s in that fi n ite time p erio d. Step 1 : T his ste p sho ws that i f an agen t do es n ot r eceiv e new i nformation from its out-neigh b ors, its inte r-ev en ts time is b ound ed b y a p ositiv e constan t. Assume that agen t i ∈ V has ju st b roadcast its state at time t 0 , then e i ( t 0 ) = 0. F or t > t 0 , wh ile no n ew in f ormation is recei v ed, ˆ x i ( t ) and ˆ x j ( t ) remain unc h anged. Give n that ˙ e i = − ˙ x i , the ev olution of the error is simply e i ( t ) = − ( t − t 0 ) ˆ z i , (21) where ˆ z i = P j ∈N out i w ij ( ˆ x j − ˆ x i ). Since w e are considering the case that no neigh b ors of agen t i broadcast their states, therefore trigger (19) is irrelev ant. W e then n eed to find out the next time p oint t ∗ when f i ( e i ( t ∗ )) = 0 and agen t i is triggered to broadcast. This can b e d one follo wing trigger (18). If ˆ z i = 0, n o broadcasts will ev er happ en b ecause e i ( t ) = 0 for all t ≥ t 0 . Consider the case when ˆ z i 6 = 0, u s ing (21), 8 trigger (18) prescrib es a broadcast at time t ∗ ≥ t 0 that satisfies ( t ∗ − t 0 ) 2 ˆ z 2 i − 2 σ i δ i b i c i ( b i + c i ) d out i ˆ z 2 i = 0 , or equiv alen tly ( t ∗ − t 0 ) 2 = 2 σ i δ i b i c i ( b i + c i ) d out i . Therefore, we ca n lo wer b ound the in ter-ev en ts time by τ i = t ∗ − t 0 = q 2 σ i δ i b i c i ( b i + c i ) d out i , whic h explains our c hoice in (20). By this step, if none of agen t i ’s neigh b ors broadcast, agen t i w ill not b e triggered infinitely fast. Next, w e sh o w that messages can not b e sent infinitely o v er a fin ite time p erio d when one or more neigh b ors of agen t i trigger(s). Step 2 : Same as Step 1 , assume agent i h as just b roadcast its state at time t 0 , th us e i ( t 0 ) = 0. Our reasoning is as follo ws: 1) If no information is receiv ed by time t 0 + ε i < t 0 + τ i , then n o trigger happ ens for agen t i . 2) Let us then consider the situation that at least one neigh b or of agen t i br oadcasts its information at some time t 1 ∈ ( t 0 , t 0 + ε i ), whic h m eans th at agen t i would also re-broadcast its information at time t 1 due to trigger (19). Define I as the set in whic h all agents h a ve broadcast information a t time t 1 , then as long as no ag en t k ∈ I send s new informatio n to any agen t in I , ag en ts in I will not broadcast new information for at lea st min j ∈ I τ j seconds, whic h includes the original agen t i . As no new information is receiv ed by an y agen t in I by time t 1 + min j ∈ I ε j , there is no problem. 3) Again consider the ca se that at least one a gen t k sen d s new informatio n to some agen t j ∈ I at time t 2 ∈ ( t 1 , t 1 + min j ∈ I ε j ), then b y trigger (19), all agen ts in I w ould also broadca st their st ate inform ation at time t 2 and agen t k will n o w be added to I . Th e remaining reasoning is just to rep eat what has b een reasoned, th us, the only situatio n for infinite comm unications to occur in a finite ti me p erio d is to h a ve a net work of infinite agen ts, which is imp ossible for the N -agen t netw ork we co nsider. Therefore, Step 1 and Ste p 2 conclude that Algorithm 2 excludes Zeno b eha vior for the net w ork. Next w e establish the global exp onential co n v ergence. Theorem 5.2. (Exp onential Conver genc e to Aver age Consensus). Given the sys- tem (3) exe cuting T able 2 over a weight-b alanc e d, str ongly c onne cte d digr aph, al l agents exp onential ly c onver ge to the aver age of their initial states, i.e. lim t →∞ x ( t ) = ¯ x , wher e ¯ x = 1 N P N i =1 x i (0) 1 N . Pr o of. The triggering ev ents (17) and (18) ensure that ˙ V 2 ( x ( t )) ≤ P N i =1 ( σ i − 1) δ i  P j ∈N out i w ij ( ˆ x i ( t ) − ˆ x j ( t ))  2 . (22) T o show that the con v ergence is exp onentia l, we sho w that the evolutio n of V 2 ( x ( t )) to wards 0 is exp onen tial. Omit the time stamp t for simp licit y , and defin e σ max = 9 max i ∈V σ i , δ max = max i ∈V δ i to further b ound (2 2): ˙ V 2 ( x ) ≤ ( σ max − 1) δ max N X i =1  X j ∈N out i w ij ( ˆ x i − ˆ x j )  2 = ( σ max − 1) δ max ˆ x T L T L ˆ x ≤ ( σ max − 1) δ max λ 2 ( L ) ˆ x T L T ˆ x, where w e use (1) to come up with the last inequalit y . Note that V 2 ( x ) = 1 2 x T L T x = 1 2 ( ˆ x − e ) T L T ( ˆ x − e ) = 1 2 ( ˆ x T L T ˆ x − ˆ x T L T e − e T L T ˆ x + e T L T e ) ≤ 1 2 (2 ˆ x T L T ˆ x + 2 e T L T e ) ≤ ˆ x T L T ˆ x + k L kk e k 2 . (23) Substitute (15) in to (23), define d out min = min i ∈V d out i , 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 ha ve ˆ x T L T ˆ x + k L kk e k 2 ≤ ˆ x T L T ˆ x + k L k 2 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 2 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) giv es ˙ V 2 ( x ) ≤ ( σ max − 1) δ max λ 2 ( L ) ˆ x T L T ˆ x ≤ ( σ max − 1) δ max λ 2 ( L ) 2(1 + 2 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 − 1)( b min + c min ) δ max λ 2 ( 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 2 ( x ) . (25) Substitute A = ( σ max − 1)( b min + c min ) δ max λ 2 ( 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 ) in to (25), we ha v e ˙ V 2 ( x ( t )) ≤ AV 2 ( x ( t )), therefore we conclude that V 2 ( x ( t )) ≤ V 2 ( x (0)) exp ( At ) and the net work con verges exp onential ly to the av erage of its initial s tate. With the theoretical foundation of Algo rithm 2 , we are no w ready to prop ose a class of ev ent- triggered al gorithms that can b e tu ned to meet v arying p erf orm ance needs under different scenario s. 10 6. A Class of Ev en t -T riggered Algorithms As stat ed in Sect ion 1, f or a giv en sys tem, th er e are man y w orks studyin g ev ent - triggered control using Lya pun ov fu n ctions to reac h the goal of maintaining the s ta- bilit y of the system, while increasing th e e fficiency of the system. H o wev er, there is very little w ork cur ren tly a v ailable that mathematically quan tifies th ese b enefits. Recen tly , some wo rks b egan establishing results along this line (Antunes & Heemel s, 2014; Khasho o ei et al., 2017; Ramesh et al., 2016), still this area is in its infancy . In particular, there are n ot yet established wa ys to compare the p erformance of an ev ent-trig gered algorithm with another. Consequ en tly , many differen t algorithms can b e prop osed to ultimately s olv e the same problem, while eac h a lgorithm is sligh tly dif- feren t and pro du ces differen t tra jectories. Sp ecifically in ou r case, Algorithm 1 and Algorithm 2 solv e the exact same p roblem, and offer the exact same guarante es, i.e., they b oth exclude Zeno b ehavio r and ensu r e asymptotic conv ergence of the n et work. So, whic h algorithm should we use? Moreo ve r, we ha v e found that dep ending on the initial conditions and n etw ork top ology , eac h algorithm ma y out-p erform the other in terms of different ev aluation metrics. In any case, once these p erf ormance metrics b ecome b etter researc hed, there will lik ely b e more standard w a ys to mathematically compare the tw o d ifferen t algorithms. Th er efore, f or no w, instead of d esigning only one eve nt -triggered algorithm for the system that only works b etter in one situation, w e aim to d esign an ent ire class of algorithms th at can easily b e tuned to meet v arying p erformance needs. W e do th is by p arameterizing a set of Ly apuno v functions rather than studyin g only a sp ecific one. T o the b est of our knowledge, this pap er is then a first study of ho w to design an ent ire class of algorithms th at use different Ly apuno v functions to guaran tee correctness, with the in tention of b eing able to u se th e b est one at all times. In this pap er, we utilize only tw o Lyapuno v functions, ho w ev er, w e can also use as man y Ly ap u no v functions as we wa n t and com b ine th em all to deve lop the en tire cl ass of algorithms. Sp ecifically , giv en any λ ∈ [0 , 1], w e define a com b ined Lya punov function as V λ ( x ( t )) = λV 1 ( x ( t )) + (1 − λ ) V 2 ( x ( t )) . (26) Accordingly , the deriv ative o f V λ ( x ( t )) tak es the form ˙ V λ ( x ( t )) = λ ˙ V 1 ( x ( t )) + (1 − λ ) ˙ V 2 ( x ( t )) . (27) F ollo wing th e steps of deriving the triggering fun ctions in Section 4, the triggering function dev elop ed based on the com bined Ly apuno v function (26) is giv en by f i ( e i ( t )) = e 2 i ( t ) − σ i h λ 4 d out i X j ∈N out i w ij  ˆ x i ( t ) − ˆ x j ( t )  2 + (1 − λ )2 δ i b i c i ( b i + c i ) d out i  X j ∈N out i w ij ( ˆ x i ( t ) − ˆ x j ( t ))  2 i . (28) W e refer to the algorithm d ev elop ed f rom the com bined Lyapuno v fu nction as the Com bined Algorithm parameterize d b y λ , w ith λ ∈ [0 , 1]. Note that λ = 0 reco v ers Algorithm 2 and λ = 1 reco v ers Algorithm 1 . 11 Similarly , for the Com bine d Algorithm , w e use the follo wing ev ents to a void missing any tr iggers: f i ( e i ( t )) > 0 , (29) f i ( e i ( t )) = 0 an d φ i 6 = 0 , (30) where, with a slight abuse of n otation, φ i = λ 4 d out i P j ∈N out i w ij ( ˆ x i ( t ) − ˆ x j ( t )) 2 + (1 − λ )2 δ i b i c i ( b i + c i ) d out i ( P j ∈N out i w ij ( ˆ x i ( t ) − ˆ x j ( t ))) 2 . The parameter that b oun ds the in ter-ev en ts time and excludes Z eno b eha vior is also designed: ε i < q λσ i 4 d out i w max i |N out i | + 2(1 − λ ) σ i δ i b i c i ( b i + c i ) d out i . Then, with the triggering function (28) and ε i defined ab o ve, th e C om bined Al- gorithm can also b e implemente d us ing T able 2. Corollary 6.1. Both Algorithm 1 and A lgorithm 2 ensur e al l agents to exp onen- tial ly c onver ge to the aver age of their initial states with the pr o of that their Lyapunov functions c onver ge exp onential ly. Ther efor e, as a line ar c ombination of V 1 ( x ( t )) and V 2 ( x ( t )) , V λ ( x ( t )) also c onver ges exp onential ly, which me ans that a network exe cuting the Combine d Algorithm shal l c onver ge exp onential ly to the aver age of its initial states. T o illustrate the c orrectness a nd effectiv eness of Algorithm 2 and the Com bined Algorithm , w e int ro du ce the fundamen tal clo ck sy n c h ronization problem that exists in wireless sensor net w orks (WSNs) as a case stu dy . 7. Case St udy: C lo c k Sync hronization 7.1. Backgr ound WSNs are broadly app lied in areas such as disaster managemen t, b order protection, and securit y s u rv eillance, to name a few, thanks to their lo w-cost and collaborative na- ture (Abbasi & Y ounis, 2007; Gu n gor, Lu, & Hanc ke, 2010 ). How ev er, the underlying lo cal clo c ks of these sensors are often in d isagreemen t due to the imp erfections of clo c k oscillato rs. T o guaran tee consistency in the co llected data, it is crucial to syn c h r onize these clocks w ith high p recision. In add ition, as the s mall m icro-pro cessors em b edded in eac h sensor no de are us u ally resour ce-limited (Gungor et al., 2010), energy-efficien t comm u nication proto cols for clo c k synchronizatio n are therefore desired. Quite a lot appr oac hes h av e b een p rop osed to s olv e this problem, ranging from cen- tralized to distributed, time-triggered to ev ent-trigg ered, see ( Carli & Zampieri, 201 4; Chen, Li, Huang, & T ang, 2015; Choi & Shen, 2010; Garcia , Mou, Cao, & Casb eer, 2017; Kado waki & Ishii, 2015; Mar´ oti, Kusy , Simon, & L ´ edeczi, 2004; Simeone & Spagnolini, 2007; Solis, Bork ar, & Kumar, 20 06) and references therein. T o solv e this fundament al problem, we pr op ose to apply our ev en t-triggered algo- rithms, i.e ., Algorithm 2 a nd the C om bined Algorithm in this practica l case. One of th e most related works is done by Chen et al. (2015), wh ere an ev en t-triggered algorithm with st ate-dep endent triggers is prop osed. Ho w ever, the virtual clo c ks they sync hronize are formed in a discrete man n er, which ma y encounter abrupt changes. 12 The a bilit y of a voiding a brup t c h anges is essent ial in cl o c k sync h ronization s in ce time discon tin u it y due to these c hanges can cause serious faults such as m issing imp ortant ev ents (Su ndararaman, Buy , & Kshemk aly ani, 2005). While another ev ent- triggered algorithm pr op osed by Garcia et al. (2017) do es synchronize con tinuous-time virtual clocks, h o wev er, their time-dep endent trigger design requ ires global information. Motiv ated by these tw o works, w e introd u ce our state-dep endent even t-triggered algorithms that syn chronize co n tin uous-time vir tual clocks. 7.2. Clo ck synchr onization pr oblem form ulation Consider an N -sensor W SN whose top ology is d escrib ed b y a strongl y-connected w eigh t-balanced und erlying digraph G = {V , E , W } , with V , E , W defi n ed as in Sec- tion 2. Without loss of ge neralit y , w e sa y that a se nsor i is a ble to receiv e information from its neigh b ors in N out i and send in formation to n eigh b ors in N in i . Eac h sens or in the net w ork is equipp ed with a micropro cessor with an und erlying lo cal cloc k l i ( t ), whic h is a fun ction of the abs olute time t ∈ R ≥ 0 . I deally , the lo cal clo c ks sh ould b e configured as l i ( t ) = t so that the not ion of time is consisten t through ou t the system. In realit y (Kado w aki & Ishii, 2015), how ev er, they are in the form of l i ( t ) = γ i t + o i , i = 1 , . . . , N , (31) where the unknown constan ts γ i ∈ R > 0 and o i ∈ R represen t the cloc k drift and offset of i -th clo c k, resp ectiv ely . As the absolute time t is n ot av ailable, the clock d rift γ i and offset o i can n ot b e computed directly . T o sync hronize the system, here w e mean to sync hronize the virtual clocks T i ( t ) of all sensors defin ed b y (Kado w aki & Ish ii, 2015) T i ( t ) = α i ( l i ( t )) l i ( t ) , i = 1 , . . . , N , (32) where α i ( l i ( t )) is the con trolled drift and is a fun ction of no de i ’s local time l i ( t ). The clo c k synchronizat ion is said to b e ac hiev ed if lim t →∞ | T i ( t ) − T j ( t ) | = 0 , ∀ i, j ∈ { 1 , . . . , N } . (33) F or sim p le implementa tion, in this pap er we consider the p articular case where only clock dr ift is presen t, i.e., the clo ck offset o i = 0 f or i = 1 , . . . , N . W e also assum e γ i ∈ [1 − ǫ γ , 1 + ǫ γ ], where ǫ γ is kno wn. The lo cal clo c ks are then give n by l i ( t ) = γ i t, i = 1 , . . . , N . (34) Substitute (34) int o (32) giv es the exp ressions of virtual clocks T i ( t ) = γ i α i ( l i ( t )) t, i = 1 , . . . , N . (35) Note t hat the virtual clo cks are con tin uous b y definition, t herefore the abrupt c h an ges on the clo c ks are a v oided. The dynamics of α i ( l i ( t )) is sp ecified by d α i ( l i ( t )) d l i ( t ) = − P j ∈N out i w ij ( ˆ α i ( l i ( t )) − γ j γ i ˆ α j ( l j ( t ))) , (36) 13 where ˆ α i ( l i ( t )), ˆ α j ( l j ( t )) represent the last broadcast state v alues of sensor i and j at their lo cal time l i and l j , resp ectiv ely . T h ough γ i and γ j can n ot b e computed directly , the v alue of γ j γ i can b e obtained as follo ws (Garcia et al., 2017): record the lo cal time of no de i and n o de j wh en no d e i receiv es i nformation fr om no de j at t w o time p oin ts, sa y t m and t n , th en a j a i can b e computed using γ j γ i = l j ( t m ) − l j ( t n ) l i ( t m ) − l i ( t n ) . Note we on ly n eed the lo cal clo c k time, not the exact v alues 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, wher e α i ( l i ( t )) is its current con trolled drift. An ev en t for s ensor i is triggered as so on as th e triggering function f i ( l i ( t ) , α i ( l i ( t )) , ˆ α i ( l i ( t )) , ˆ α j ( l j ( t ))) > 0 (37) is fulfilled. The triggered ev ent then driv es sensor i to broadcast its cu r ren t stat e α i ( l i ( t )) to its neighbors so that they can up date their states accordingly . Our ob jectiv e is to apply Algorithm 2 and th e Combined Algorithm so as to design triggering functions (3 7) for eac h sensor with its lo cally a v ailable information so that the virtu al clocks are synchronized, i.e ., (33) is satisfied. 7.3. Distribute d event-trigg er e d clo ck synchr onization a lgorithms The e v en t-triggered al gorithms for clo c k synchronizatio n are dev elop ed based on Lya- punov f unctions. T o b egin, 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 mo dified dr ift. It is clear that once consensus is ac hieve d on the v ariables y i ( t ), t he cl o c k synchronizati on will b e realized regardless of the individu al v alues of γ i and α i ( l i ( t )). W e th en adopt the Ly apunov cand id ate functions prop osed in Sect ion 4, with the mo d ified drifts as v ariables, i. e., V 1 ( y ( t )) = 1 2 ( y ( t ) − ¯ y ) T ( y ( t ) − ¯ y ), V 2 ( y ( t )) = 1 2 y ( t ) T L T y ( t ), and V λ ( y ( t )) = λV 1 ( y ( t )) + (1 − λ ) V 2 ( y ( t )). As the algorithm de- v elopment with differen t Ly apun o v fun ctions are simila r, w e o nly use V 2 ( y ( t )) = 1 2 y ( t ) T L T y ( t ) as an example to illustrate the d eriv ation pro cess. The dynamics of the mo dified drift y i ( t ) is deriv ed as follo ws: ˙ y i ( t ) = d α i ( l i ( t )) d l i ( t ) · d l i ( t ) d t = γ 2 i  − X j ∈N out i w ij ( ˆ α i ( l i ( t )) − γ j γ i ˆ α j ( l j ( t )))  = − γ i  X j ∈N out i w ij ( γ i ˆ α i ( l i ( t )) − γ j ˆ α j ( l j ( t ))  = − γ i X j ∈N out i w ij  ˆ y i ( t ) − ˆ y j ( t )  . (39) W e then sp ecify the foll o w ing Lemma to u pp er b ound the deriv ativ es of V 2 ( y ( t )). Lemma 7.1. In clo ck synchr onization, for i ∈ V , let b i , c j > 0 for al l i, j ∈ V (the same b i , c j as in L e mma 4.1, define ν i ( t ) = P j ∈N out i w ij ( ˆ y i ( t ) − ˆ y j ( t )) , and e y i ( t ) = 14 ˆ y i ( t ) − y i ( t ) , then the derivative of V 2 ( y ( t )) = 1 2 y ( t ) T L T y ( t ) is upp er b ounde d by ˙ V 2 ( y ( t )) ≤ − P N i =1 γ i  δ i  P j ∈N out i w ij ( ˆ y i ( t ) − ˆ y j ( t ))  2 −  d out i 2 b i + d out i 2 c i  e 2 y i ( t )  , (40) wher e δ i is what define d i n (14) . The pro of is similar to the pro of for Lemma 4.1 and is omitted d ue to space limit. F r om Lemma 7.1, w e can see that as long as ˙ V 2 ( y ( t )) < 0 and k Ly ( t ) k 6 = 0 hold, y i ( t ) ac hieves 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 sync hronization on virtual clocks can b e ac h iev ed. A sufficien t condition to ensure that V 2 ( y ( t )) is monotonically decreasing is e 2 y i ( t ) ≤ δ i ( d out i 2 b i + d out i 2 c i )  X j ∈N out i w ij ( ˆ y i ( t ) − ˆ y j ( t ))  2 = 2 b i c i δ i ( b i + c i ) d out i  X j ∈N out i w ij ( γ i ˆ α i ( l i ( t )) − γ j ˆ α j ( l i ( t )))  2 = 2 γ 2 i b i c i δ i ( b i + c i ) d out i  X j ∈N out i w ij ( ˆ α i ( l i ( t )) − γ j γ i ˆ α j ( l j ( t )))  2 . (41) With e y i ( t ) = γ i e i ( l i ( t )), w e define the triggering fu nction dev elop ed from Algo- rithm 2 as f i ( e i ( l i ( t ))) = e 2 i ( l i ( t )) − 2 b i c i δ i ( b i + c i ) d out i  X j ∈N out i w ij ( ˆ α i ( l i ( t )) − γ j γ i ˆ α j ( l j ( t )))  2 . (42) T o ensure no triggers are missed by sensor i , w e defi ne an ev ent eit her by f i ( e i ( l i ( t ))) > 0 or (43) f i ( e i ( l i ( t ))) = 0 and X j ∈N out i w ij ( ˆ α i ( l i ( t )) − γ j γ i ˆ α j ( l j ( t ))) 6 = 0 . (44) Similarly , an a dditional trigger is p r escrib ed to address the n on-Zeno b eha vior. Let l last i b e the last time at whic h sensor i broadcasts its in f ormation to its neigh b ors. If at some time l i ( t ) ≥ l last i , sensor i receiv es information fr om a neighbor j ∈ N out i , then it immediately broadcasts its state if l i ( t ) ∈ ( l last i , l last i + ε ′ i ) , (45) where ε ′ i < s 2 σ i b i c i δ i ( b i + c i ) d out i (46) whose design is as giv en in Prop osition 5.1. The follo wing result presen ts Algorithm 2 in the cloc k sync h ronization applica tion. 15 Theorem 7.2. F or an N -sensor network over a weight-b alanc e d digr aph, assume only clo ck drift exists, i.e., o i = 0 , ∀ i ∈ V . With the v i rtual clo cks (32) , dynamics given in (36) , the distribute d event-trigger e d c onsensus algorithm (42) - (46) ( Algorithm 2 ) achieves asymptotic synchr onization for the virtual clo cks, i.e., (33) is satisfie d. W e ha ven sho w n that Algorithm 2 can b e applied to the practical clo ck sync hro- nization problem. Next, we show that the Combined Algorithm can also b e applied to solv e the clo ck sync hronization problem. T o d o so, we first deriv e the triggering la w for the clo c k synchronizatio n pr oblem from Algorithm 1 as f i ( e i ( l i ( t ))) = e 2 i ( l i ( t )) − σ i 4 d out i X j ∈N out i w ij ( ˆ α i ( l i ( t )) − γ j γ i ˆ α j ( l j ( t ))) 2 , (47) with an int er-ev ent p erio d b ounded b y ε ′ i < q σ i 4 d out i w max i |N out i | . Then, with the triggering rules (42) - (47) and the analysis in Section 6, d esigning the triggering function for the clo c k synchronizat ion pr ob lem from the Com bined Algorithm is straigh tforw ard. Th at is, f i ( e i ( l i ( t ))) = e 2 i ( l i ( t )) − λσ i 4 d out i X j ∈N out i w ij ( ˆ α i ( l i ( t )) − γ j γ i ˆ α j ( l j ( t ))) 2 − 2(1 − λ ) σ i δ i b i c i ( b i + c i ) d out i  X j ∈N out i w ij ( ˆ α i ( l i ) − γ j γ i ˆ α j ( l j ))  2 , (48) with an int er-ev ent p erio d b ounded b y ε ′ i < q λσ i 4 d out i w max i |N out i | + 2(1 − λ ) σ i δ i b i c i ( b i + c i ) d out i . Theorem 7.3. F or an N -sensor network over a weight-b alanc e d digr aph, assume only clo ck drift exists, i.e., o i = 0 , ∀ i ∈ V . With the v i rtual clo cks (32) , dynamics given in (36) , the d istribute d event trigging rule define d in (48) , then the Combin e d A lgo- rithm achieves asymptotic synchr onization for the v i rtual clo cks when the triggering c ondition f i ( e i ( l i ( t ))) > 0 or f i ( e i ( l i ( t ))) = 0 with e 2 i ( l i ( t )) > 0 is met. The pr o of of the theorem and the stabilit y analysis, non-Zeno b eha vior exclusion are as giv en in Section 5, therefore are omitted. 8. Sim ulat ion Results In this section, w e a pply Algorithm 1 and Algorithm 2 to the ev ent-trigg ered clock sync h ronization problem, to sho w the effectiv eness of b oth a lgorithms. W e then demonstrate the p er f ormance of the prop osed algorithms through sev eral simulatio ns and sh o w h o w either Algorithm 1 or Algorithm 2 could b e argued to b e ‘b etter’ giv en differen t netw ork top ology , which has set the basis for our introd uction of th e Com bined Algorithm to easily go b etw een the tw o. W e first sh o w th at b oth Algorithm 1 and Algorithm 2 are able to syn c hr onize the virtu al clocks in WSNs. W e consider four d ifferen t net w ork top ologie s, with th eir corresp onding weig h ted adjacency m atrices listed in T able 3. The clock offset is 0 for all no des, and the unkn o wn clo c k drifts are γ = [0 . 65 0 . 79 0 . 91 1 . 25 1 . 4] T . The ev olution of the lo cal clo c ks with resp ect to the absolute 16 T able 3. F our different net works. Net w ork 1: Random netw ork Net w ork 2: Ring netw ork W 1 =       0 1 0 0 0 0 0 1 / 2 1 / 2 0 5 / 6 0 1 / 6 0 0 1 / 6 0 1 / 6 1 / 2 1 / 6 0 0 1 / 6 0 5 / 6       W 2 =       0 1 / 2 0 0 1 / 2 1 / 2 0 1 / 2 0 0 0 1 / 2 0 1 / 2 0 0 0 1 / 2 0 1 / 2 1 / 2 0 0 1 / 2 0       Net w ork 3: C omplete netw ork Net w ork 4: Star netw ork W 3 =       0 1 / 4 1 / 4 1 / 4 1 / 4 1 / 4 0 1 / 4 1 / 4 1 / 4 1 / 4 1 / 4 0 1 / 4 1 / 4 1 / 4 1 / 4 1 / 4 0 1 / 4 1 / 4 1 / 4 1 / 4 1 / 4 0       W 4 =       0 1 / 4 1 / 4 1 / 4 1 / 4 1 / 4 0 0 0 0 1 / 4 0 0 0 0 1 / 4 0 0 0 0 1 / 4 0 0 0 0       0 5 10 15 20 0 5 10 15 20 25 30 l ( t ) t (a) Local clo cks 0 5 10 15 20 0 10 20 0 5 10 15 20 0 10 20 T ( t ) T ( t ) t (b) Virtual clocks Figure 1. Plots of the sim ulation results of th e clo c k synchroniza tion on Netw ork 1. (a) The lo cal clocks are the same for both algori thms. (b) Virtual cl o cks with the implemen tation of ev en t-triggered control. Both Algorithm 1 (top) and Algori thm 2 (bottom) ar e able to sync hronize the vir tual clocks. time t is sho w n in Figure 1a. W e can see that without an y control, the lo cal clo c ks will div erge. Then, we implemen t Algorithm 1 and Algorithm 2 with the con trol la w (36), triggering fu n ctions (47) and (42 ) dev elop ed from Algorithm 1 and Algorithm 2 , resp ectiv ely , to achiev e clock sync hronization. Th e inv olv ed parameters are set to b e σ i = 0 . 5, and b i = c i = 0 . 5 /d out i for all i ∈ { 1 , . . . , N } . Both algorithms are able to sync hronize the virtual clo c ks on all four net w orks and we tak e the resu lt on Netw ork 1 as an example and show th e v ir tual clo c k evolutio n in Figure 1b. Ho wev er, except for the sync h ronization, w e h a ve n o idea wh ich algorithm p erforms b etter on other ev aluation metrics, f or example, th e con vergence sp eed and total energy consu mption. Also, the p erformance ev aluation result ma y differ for different netw ork top ologies. Therefore, in the follo win g sim ulations, we show the difference of the t w o different algorithms on four netw ork top ologies with different ev aluation metrics. W e plot the triggering instances of all no des in the net w ork when implementi ng the ev en t-triggered algorithms in Figure 2a, 2c, 2e, 2g. W e notice that in general, the n umber of ev ents triggered when implementing Algorithm 1 is less than that when implemen ting Algorithm 2 . W e also p lot the evo lution of Ly apuno v functions, i.e., V 1 ( y ( t )) = 1 2 ( y ( t ) − ¯ y ) T ( y ( t ) − ¯ y ), V 2 ( y ( t )) = 1 2 y ( t ) T L T y ( t ), and V λ ( y ( t )) = λV 1 ( y ( t )) + 17 (1 − λ ) V 2 ( y ( t )) for all net w orks w ith σ i = 0 . 5 for all agen ts in Figure 2b, 2d, 2f, 2h, whic h again corrob orates o ur analysis that b oth algorithms ensure con v er gence, or in this case, sync h ronization for the resulting s y s tems. W e can see that except for Net w ork 3 , the Lyapuno v function of Algorithm 2 in the other three net wo rks con verges faster than that of Algo rithm 1 . It is also noted that when the n umber of even ts triggered wh en implemen ting Algorithm 2 is n oticeably larger than that in implemen ting Algorithm 1 , th e conv ergence sp eed of the Lyapuno v fun ction in Algorithm 2 is also n oticeably faster than that in Algorithm 1 . This is reasonable, since the more even ts are triggered, the more informatio n is comm unicated in the net work, and the faster the consensus will b e reac hed. The ab o ve sim ulations corrob orate our argumen t that dep ending on the c hosen ev aluation metric, either algorithm can b e argued to b e ‘b etter’ than the other. T o b etter quanti ze/visualize the p erformance difference of the t wo algorithms and demon- strate our motiv ations for pr op osing the Combined Algorithm , we then executing all algorithms with resp ect to v arying σ . W e ev aluate th ese algorithms with four p er- formance metrics, 1) the t otal num b er of ev ent s t riggered, denoted b y N e , 2) the time needed for eac h net w ork to reac h a 99% con verge nce of the Ly apunov fun ction, denoted by T con , 3) the total comm unication energy required to achiev e a 99% con- v er gence, denoted b y E , and 4) the square of the H 2 -norm of the system, denoted b y C (Dezfulian, Ghaedsharaf, & Motee, 2018). The total comm unication energy n eeded is calculated by multiplying the p o w er in un its of milli w att (mW) with T con , where we adopt the follo wing p ow er calculation mo del in u nits of mW (Ma rtins et al. , 2008): P = N X i =1   X j ∈{ 1 ,. ..,N } ,j 6 = i η 10 0 . 1 P i → j + ζ k α i ( l i ( t )) − α j ( l j ( t )) k   , where ζ > 0 and η > 0 dep end on the c haracteristics of the wireless medium an d P i → j is the p o wer of the s ignal transmitted fr om agen t i to agen t j in units of dBmW. Similar as (No wzari & Cort ´ es, 2012), we set η , ζ and P i → j to b e 1. The square of t he H 2 -norm, C is defined by C := Z ∞ t =0 N X i =1 ( y i ( t ) − ¯ y ) 2 dt, where y i ( t ) is the m o dified drif t of eac h lo cal clo ck and ¯ y is the a v erage of th e mo d ified drift of the system. The inv olv ed parameters are set to b e b i = c i = 0 . 5 /d out i for all i ∈ { 1 , . . . , N } . The same cont rol law (36) is ap p lied. F or Algorithm 1 and Algorithm 2 that ac hieve clo c k synchronizatio n, their triggering functions are giv en by (47) and (42), resp ectiv ely . F or the Com bined Algorithm , its triggering function is giv en b y (48), with λ = 0 . 5. F or eac h σ , w e ru n 10 sim ulations with r andom clo ck d rift that satisfies to γ i ∈ (0 . 7 , 1 . 3) and obtain the a verage of N e , T con , E , and C in eac h simulation. F r om the top figu r es in Figure 3a, 3c, 3e, and 3g , w e can see that for d ifferent σ , th e total num b er of ev ents triggered in the system when executing Algorithm 2 is larger t han that w hen executing Alg orithm 1 . On the other h and, from the b ottom figures in Figure 3a, 3c, 3e, and 3g, we ca n see that th e time needed to reac h a 99% con verge nce of the system is u s ually m uc h less wh en executing Algorithm 2 , with the only exception for the c omplete net w ork, w h ere b oth algorithms ha v e similar 18 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1415 1617 18 19 20 1 2 3 4 5 0 1 2 3 4 5 6 7 8 9 10 1112 13 14 15 16 17 18 19 20 1 2 3 4 5 Ev ent s Ev ent s t (a) Net work 1, triggering i nstances 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 1.2 Algorithm 1 Algorithm 2 V ( t ) t (b) Net work 1, Ly apunov function 0 1 2 3 4 5 6 7 1 2 3 4 5 0 1 2 3 4 5 6 7 1 2 3 4 5 Ev ent s Ev ent s t (c) Net work 2, triggering i nstances 0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 Algorithm 1 Algorithm 2 V ( t ) t (d) Net work 2, Ly apunov function 0 1 2 3 4 1 2 3 4 5 0 1 2 3 4 1 2 3 4 5 Ev ent s Ev ent s t (e) Net work 3, triggering i nstances 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 1.2 Algorithm 1 Algorithm 2 V ( t ) t (f ) Netw ork 3, Lyapuno v function 0 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 0 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 Ev ent s Ev ent s t (g) Net work 4, triggering i nstances 0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 Algorithm 1 Algorithm 2 V ( t ) t (h) Net work 4, Ly apunov function Figure 2. Pl ots of the triggering instances and the evolution of Lyapuno v candidate functions on four netw orks when implemen ting both Algorithms. F or figure (a) , (c), (e), (g), Algo rithm 1 is on the top and Algorithm 2 is on the bottom. 19 0 0.2 0.4 0.6 0.8 1 10 3 Algorithm 1 Algorithm 2 Combined Algorithm 0 0.2 0.4 0.6 0.8 1 4 5 6 7 N e T con σ (a) Net work 1 0 0.2 0.4 0.6 0.8 1 4 5 6 7 0 0.2 0.4 0.6 0.8 1 4.2 4.4 4.6 10 -5 E C σ (b) Net work 1 0 0.2 0.4 0.6 0.8 1 10 3 Algorithm 1 Algorithm 2 Combined Algorithm 0 0.2 0.4 0.6 0.8 1 1.6 1.8 2 2.2 2.4 N e T con σ (c) Net work 2 0 0.2 0.4 0.6 0.8 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 1.6 1.8 2 10 -5 E C σ (d) Net work 2 0 0.2 0.4 0.6 0.8 1 10 3 Algorithm 1 Algorithm 2 Combined Algorithm 0 0.2 0.4 0.6 0.8 1 1 1.2 1.4 1.6 N e T con σ (e) Net work 3 0 0.2 0.4 0.6 0.8 1 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.6 1.8 2 2.2 10 -5 E C σ (f ) Net wo rk 3 0 0.2 0.4 0.6 0.8 1 10 2 Algorithm 1 Algorithm 2 Combined Algorithm 0 0.2 0.4 0.6 0.8 1 4 6 8 N e T con σ (g) Net work 4 0 0.2 0.4 0.6 0.8 1 4 6 8 0 0.2 0.4 0.6 0.8 1 6 7 10 -5 E C σ (h) Net work 4 Figure 3. Plots of different ev aluation metrics. F or figure (a), (c), (e), (g), top: total eve nt s triggered, b ottom: con v ergence time (bottom); for figure (b), (d), (f ), (h), top: energy consumption, bottom: H 2 -norm squared. 20 con vergence sp eed. As the total comm u n ication energy consumption is related with b oth the to tal num b er of eve nt s triggered and th e ti me required to reac h conv ergence, w e can see f r om the t op figures in Figure 3b, 3d, 3f, and 3h that either algorithm can outp erform the other in terms of the total energy consu mption for different net w ork top ologies. Th e H 2 -norm squared ev aluates th e distance of eac h lo cal mo difi ed dr ift with t he av erage modifi ed drift of the system, whose v alue therefore also in dicates the con vergence sp eed of the system to some exten t, see the b ottom figures in Figure 3b, 3d, 3f, and 3h. Therefore, dep ending on different net work top ologies and dep ending on what p erform ance metrics are most imp ortant fo r the application at hand, it may b e desirable to implemen t differen t t yp es of ev ent- triggered a lgorithms. Note that the Com bined Algorithm can easily b e tuned to appr oac h either Algorithm 1 or Algorithm 2 or a n ything in betw een to m eet v arying system needs by setting v alues for λ . This also motiv ates our future work of adapting λ online to further impro ve p erformance. 9. Conclusion This pap er p r op oses a class o f distributed ev en t-triggered comm u nication and cont rol la w for multi-a gen t systems w hose underlying d irected graphs are weig ht -balanced. The class of alg orithms are dev elop ed fr om a class of Ly apunov fu nctions, eac h of whic h is a linear com b ination (p arameterized b y λ ∈ [0 , 1]) of t wo Ly apu n o v func- tions. Eac h λ defin es a new Ly apuno v f u nction coupled with a new even t-triggered co ordination algorithm wh ic h uses th at particular function to gu arantee correctness and is able to exclude the p ossibility of Z eno b ehavio r. W e sho w that the p rop osed en tire class of even t-triggered algorithms can b e tuned to meet v aryin g p erform ance needs b y adjusting λ . W e also apply th e pr op osed distributed even t-triggered algo- rithms to solv e the pr actical clo c k sync hronization problem in WSNs. F or the futur e researc h, we will focus on d ev eloping a unified ev aluation metric (which is a function of differen t p erf ormance n eeds) that can b e us ed to ev aluate t he p erformance of d ifferen t algorithms. In that wa y , the class of distributed algorithms will b e dev elop ed from a tunable algorithm to an adaptiv e algorithm. F unding This w ork w as supp orted b y the NS F under Gran t # 20429 4. References Abbasi, A. A., & Y ounis, M. (2007 ). 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(A1) 23 Substitute the ve ctor form x = ˆ x − e into (A1), and expand it with (3), w e ha ve ˙ V 2 ( x ) = ˆ x T L T ˙ x − e T L T ˙ x = N X i =1 ( X j ∈N out i w ij ( ˆ x i − ˆ x j ) u i − X j ∈N out i w ij ( e i − e j ) u i ) = N X i =1 ( − u 2 i − X j ∈N out i w ij e i u i + X j ∈N out i w ij e j u i ) = N X i =1 ( − u 2 i − d out i e i u i + X j ∈N out i w ij e j u i ) . (A2) F or b i , c j > 0, ap p lyY oung’s inequalit y (2) to the cross terms at the right hand side of (A2) giv es − d out i e i u i ≤ d out i 2 b i e 2 i + d out i b i 2 u 2 i , X j ∈N out i w ij e j u i ≤ X j ∈N out i w ij 2 c j e 2 j + X j ∈N out i w ij c j 2 u 2 i . Since the digraph is we igh t-balanced, the follo wing equalit y holds: N X i =1 X j ∈N out i w ij 2 c j e 2 j = N X i =1 X j ∈N in i w j i 2 c i e 2 i = N X i =1 d in i 2 c i e 2 i = N X i =1 d out i 2 c i e 2 i . Com bine the ab o v e inequalities and equalit y , w e obtain an upp er boun d for ˙ V 2 ( x ): ˙ V 2 ( x ) ≤ N X i =1  − u 2 i + d out i e 2 i 2 b i + d out i b i u 2 i 2 + d out i e 2 i 2 c i + X j ∈N out i w ij c j 2 u 2 i  = − N X i =1 "  1 − d out i b i 2 − X j ∈N out i w ij c j 2  u 2 i −  d out i 2 b i + d out i 2 c i  e 2 i # = − N X i =1 " δ i u 2 i −  d out i 2 b i + d out i 2 c i  e 2 i # , (A3) with δ i defined in (14). T o ensu re δ i > 0, we require b i , c j < 1 d out i . 24