Stability of Open Multi-Agent Systems and Applications to Dynamic Consensus

1 Stability of Open Multi-Agent Systems and Applications to Dynamic Consens u s Mauro Francesc helli ‡ and Paolo Frasca † Abstract In this technical note we consider a class of multi-agent network systems that we refer to as Open Multi-Agent Systems (OM AS): in these multi-agent systems, an indefinite number of agents may join or leave the netw ork at any time. Focusing on discrete-time evo lutions of scalar agents, we pro vide a novel theoretical framew ork to study t he dynamical properties of OMAS: specifically , we pro pose a suitable notion of stabilit y and deri v e sufficient conditions to ensure stability in this sense. These sufficient con ditions regard the arriv al/departure of an agen t as a disturbance: consistently , they require t he effect of arriv als/departures to be bounded (in a precise sense) and the OMAS to be contracti ve in the absence of arriv als/departures. In order to provide an exa mple of application for this theory , we re-formulate the well-kno wn Proportional Dynamic Consensus for Open Multi-Agent S ystems and we characterize the stability properties of the resulting Open Proportional Dynamic Consensus algorithm. I . I N T R O D U C T I O N A mu lti-agent system is a dy n amical model for the behavior of a possibly large grou p of agents, e.g . , robo ts, devices, sensors, oscillators etc., whose pattern of interactio ns due to sensing, commun ication o r ph ysical coupling is mo deled by a graph th at repre sents th e network struc tu re of th e system. Most literature o n mu lti-agent systems considers networks of fixed size, i.e., numb er of agents, and then conside rs several kin ds o f scenarios su c h as time-varying network topolo gies. In this p aper we explicitly c o nsider a more r a d ical scenario of open m ulti-agent system wher e the set of agents is time-varying , i. e., agents may join and leav e the network at any time. This situation is common in numero us applications, of which we mention a fe w . In the interne t of things (IoT) and smart power grids, smart de vices can join and leave the grid [1], [2]; in social network (either online or offline [3]) individuals can jo in or lea ve; in mu lti-vehicle coord ination, the com position of a robo tic team or platoon o f vehicles can ev olve with time . Despite their ubiq uity , op en m ulti-agent systems have received surprising ly little atten tion in either co n trol or in contiguo us fields. Notion s of “o pen” systems ca n be found in the computer science literature [4], [5] when referrin g to software agen ts and the pro blem of e v aluating rep utation in open environments, but not as dynamical ‡ M. Franceschelli is with the Dept. of Electrical and E lectronic Engineering, Univ ersity of Cagliari, Italy , email: mauro.franceschelli @diee.unica.it . † P . Frasca is with Univ . Grenoble Alpes, CNRS, Inria, Grenoble INP , GIPSA-lab, F-38000 Grenoble, France. email: paolo.fr asca@gipsa-lab.fr . ‡ This work was supported by the Italian Minis try of Rese arch and Education (MIUR) under c all SIR 2014 with the grant “CoNetDomeSys”, c ode RBSI14OF6H, and by Regione Autonoma della Sardegna with the 2016/2017 “Visiting Professor Program”. The research of P . Frasca was also partly supported by ANR (French National Science Fundation) via project HAND Y , number ANR-18-CE40-0010. Novem ber 27, 2024 DRAFT 2 systems. At times, dynam ically e v olving po p ulations hav e also been c o nsidered in game theory , at least to show the robustness of price-o f-anarch y results [6], [7]. Despite th e abundance of works in multi-ag ent systems from the systems and con trol commun ity , op enness is rarely explicitly includ e d in a rigorou s an alysis, b ut rather explo red by simulations as in [8]. In multi- r obot systems, where adaptivity to add ition/removal of rob ots is crucial, so m e architecture s accomm odate for dy namics team s but offer no per formanc e guarante e s [9]. Indeed, ope n ness imp lies some con ceptual difficulties in adaptin g control-theor etic notio n s such as state or stab ility . For th is reason, som e authors h av e rec e n tly p roposed to circum vent some of the m athematical hurdles b y e m bedding th e time- varying agent set in a time-inv ariant finite superset [10]. In a different persp ecti ve, others have aimed to describe the ope n multi-agen t system thro ugh significant statistical prop e rties: insightful and enco uraging results have bee n presented in [11], [1 2], where the autho rs stud y the pro blem o f av erage-co n sensus by go ssiping, and in [13], wher e th e autho rs study a m ax-consen sus problem . The co n tribution of th is pap er is twofold, as it covers both th eoretical re su lts and co n crete examples. As a theoretical contribution, we introduce an abstract framew ork for discrete-time open multi-agent s ystems: this framework is based upo n prope r definitions of state e v olutions, eq u ilibria, an d stability , and allo ws to establish useful stability criteria for a c lass of “contractive” open multi-agen t system s. In strumental to this development is extending th e notion of (Euc lid ean) distance to apply to vectors that belon g to different spaces and the r efore h ave different leng th. T h is goal is ach iev ed by our definition of op en distance fun ction. In orde r to provide a co ncrete example, we extend a distrib uted control protoc o l, namely Pro portion al Dynamic Consensus, to the open scen ario, ther e b y defining the Open Propo rtional Dynamic Consensus algorithm. Its stability proper ties can be studied by ou r analy sis tools. In th e dyn amic consen sus pr oblem , each of the node s receiv es an input sig n al and is tasked to tr ack th e average of all inpu ts over the network. Our interest in d y namic con sensus originates fro m its fu n damental r o le in distributed control in general and specifically in the do main of sma r t grid s. I n the la tter application, the object of th e distrib uted estimation ca n be the time-varying average p ower con sumption by the network. Thus, by considering the planned power co nsumption of each device as an external ref e rence signal for each ag ent, a dynamic consen sus alg orithm can be used to estimate the time-varying average value of this potentially large set of reference signals. Since devices login and log o ut from the network withou t no tice, the set of refer ence signals is, in g eneral, time - varying. The dynamic con sensus pr o blem has received sig nificant atten tion, as dem o nstrated by the forthcoming tuto - rial [14]. Since the early work in [15], a funda m ental idea to r e n der consensus proto cols “ d ynamic” has been adding the deriv ati ve o f each agents’ own ref erence sign al to a consensus filter that would thus track the time- varying average of the ref erences. Several alg orithms that exploit th is mechanism hav e b een proposed [8], [16], [17]: their main advantages are conver gence speed and accur a cy (which can be perfect for constant ref e r ence signals), while their main drawback is their lack of robustness with respect to errors in their initialization and, con sequently , with re sp ect to ch anges in th e network compo sition. If the numbe r of agents ch a nges, these alg orithms accumu la te estimation errors th a t can se verely d eteriorate the estimation perform ance. Some algorithms, for in stance those in [18] an d [19], [ 20], [2 1], have instead shown sup erior robustness pr o perties that can be useful to allow for the addition or removal of agents, even tho ugh th eir an alysis has been so far limited to n etworks of fixed size. W e Novem ber 27, 2024 DRAFT 3 take note that the ma in drawback of pr oportio n al dynam ic consensus, as illustrated in [19], consists in a trade-off between steady-state err or for constant referen ce sig nals, tracking error and co n vergence rate. In the recent work [22], [2 3], the author s propose and character ize the so- called mu lti- stag e dyna mic consensus algor ith m, which consists in a cascade of propo rtional consensus filters and has be e n proved to guaran te e sm all steady -state and tracking error for a g i ven convergence rate, thanks to exchanging a larger quantity of lo c al in formation between the agents. Further more, the strategy p roposed in [22], [23] has been shown to be implementab le with asy n chrono us and rand omized ( g ossip-based) local intera c tions. In th e rece nt confer ence publication [2 4], we gave a preliminar y account o f ou r work that con tains variations of some o f the results pr esented here . Among the m a in differences, in th e presen t paper stab ility is defined according to a normalized notion of distance , where th e d istance betwee n two states is nor malized by the square roo t of the number of agents; instead, th is norma liza tio n was no t used in [24]. Further more, th e notion of c o ntractive OMAS was not introdu ced in [24]. Structur e of the pa per: In Sec tion II we introduce the framew ork of open multi-ag ent systems (OMAS) f rom a theoretical perspectiv e and present an adaptation o f tw o k nown distributed co n trol protocols to this new framework. In Section III we p rovide theoretical tools for the stab ility analysis of discrete- tim e OMAS and app ly the resu lts to two examples of d istributed co ntrol pro to cols for O M AS. In Section V we corrob orate ou r results with numerical examples an d finally , in Section VI we d iscu ss some conclu ding rem arks. I I . O P E N DY N A M I C A L S Y S T E M S For all time k ∈ Z ≥ 0 , let G k = ( V k , E k ) be a tim e -varying directed graph with time-varying set of agen ts (a lso called nodes) V k ⊂ Z and time-varying set of edges E k ⊆ ( V k × V k ) . Set V k contains th e labels co r respond in g to the agents that are active at time k . The cardinality of set V k , that is, the n umber of agents that belong to th e network at time k , is denoted as n k = | V k | . T o av oid tri vialities, we assume that n k > 0 for all k . T w o agents v and w are said to be neighbor s at time k if they share an ed ge at time k , i.e., ( v , w ) ∈ E k . L et N v k be the set of neighbo rs of node v at time k , i.e., N v k = { w ∈ V k : ( v , w ) ∈ E k } . Let ∆ v k = | N v k | denote th e number o f neighbor s of agent v at time k . For each time k and each agent v ∈ V k , we associate a scalar state variable x v k ∈ R and an in put variable u v k ∈ R . Note th at these variables are defined on ly at time in stan ts such that v ∈ V k . More generally , in this pap er we shall call open sequ e n ce any sequ ence { y k } k where y k is indexed over V k . W ith these ingredien ts we can d e fine laws that describe how the open seq uence { x k } k ev olves. Howe ver , we will not b e able in general to write x k +1 as a function solely of x k : th e r efore, the e volution of x k does n o t constitute a “closed” dynam ical system. I n stead, we shall take as giv en the open sequences { V k } k ∈ Z ≥ 0 and { E k } k ∈ Z ≥ 0 , as well as the open sequence of inp uts { u k } k ∈ Z ≥ 0 . Provided the consistency condition s that E k ⊆ ( V k × V k ) an d u k ∈ R V k for all k , we shall define the evolution of the open sequenc e { x k } by a law x k +1 = f ( x k , V k , V k +1 , E k , E k +1 , u k , u k +1 ) . (1) Such upd ate rule should distinguish thre e k inds of nod es v , respectively belo n ging to th e sets: Novem ber 27, 2024 DRAFT 4 • R k = V k ∩ V k +1 , i.e., remaining nodes that belo ng to both V k and V k +1 ; • D k = V k \ V k +1 , i.e., de parting nodes that belo ng to V k but no t to V k +1 ; • A k = V k +1 \ V k , i.e., arriving nodes that belong to V k +1 but not to V k . Since x k must take values in R V k for all k , the compo nents corre sp onding to D k are simply lef t out f rom x k +1 . Instead, co mponen ts in A k need to b e “initialized” a ccording to some rule . Finally , for all v ∈ R k there shall be a causal evolution law in the fo r m x v k +1 = f v ( x k , V k , E k , u k ) . (2) For concreteness, we now d escribe an example of such map, which we call Open Propor tio nal Dy namic Consensus (OPDC). Dynamics 1 (Open Proportional Dynamic Consensus (OPCD)) Let ε > 0 a nd α ∈ (0 , 0 . 5) . At each time k ∈ Z ≥ 0 , each agent v ∈ V k measur es a r efer ence signal u v k and up dates its state x v k as follows: if v ∈ R k , then x v k +1 = x v k − α ( x v k − u v k ) − ε X w ∈ N v k ( x v k − x w k ); (3a) if v ∈ A k , then x v k +1 = u v k +1 . (3b) W e o b serve that if V k +1 = V k , i.e., the set of agents does not chang e , the OPDC reduces to what is called Proportio nal Dynam ic Consensus. Namely , it redu ces to the up date (3 a), which can b e written in m a tr ix form as x k +1 = x k − α ( x k − u k ) − εL k x k =  (1 − α ) I − εL k  x k + αu k = P k x k + αu k (4) where matrix P k = (1 − α ) I − εL k . W e also observe th at (3 b) n e c essarily inv olves a co m ponen t defined at time k + 1 , since u k is undefined whe n v ∈ A k . W e will study OPDC under the assum ption th at the join/leave process guaran te e s some good behavior of sequ e nce of grap h s. Assumption 2. 1 (Open Proportional Dy namic Consensus) Consider the open dynamics (3) and assume that for every k ∈ Z ≥ 0 , 1) Graph G k is undirected (that is, ( u, v ) ∈ E k if and o nly if ( v , u ) ∈ E k ); 2) max v ∈ V k ∆ v k ≤ 1 2 ε for all v ∈ V k ; 3) the number o f agents in the OMAS can not decrease too rapidly: β 2 ≤ | V k +1 | | V k | for some p ositive scalar β . W e remark that Assumption 2.1 can easily be satisfied in a distributed way and in particular does not requ ire graph G k to be co nnected for any k ∈ Z ≥ 0 . Let ¯ λ 2 ≥ 0 be a uniform lower bo und to the algebraic con n ectivity λ 2 ,L k of Novem ber 27, 2024 DRAFT 5 the Laplacian matrix L k correspo n ding to graph G k , that is, let ¯ λ 2 ≤ λ 2 ,L k . Such a constant al ways e xists (since we allow it to be zero): when it is p ositi ve, it imp lies th at all graph s are con nected and that con nectivity is unifor m ly good. I I I . S TA B I L I T Y O F O P E N M U L T I - A G E N T S Y S T E M S In our g e neral study of the stability o f OMAS, we lie down o ur instrum ents in three steps: (i) we d efine suitable (sequences o f) poin ts that play the ro le of equ ilib ria; (ii) we extend the notion o f distan ce to opera te on vecto rs of unequ a l length ; ( iii) we define a suitable notion of stab ility a n d give a sufficient conditio n fo r it. A. P oin ts of in ter est a n d their stability W e n ow define the c o ncept of trajectory of points of interest which will b e useful in the con sidered scenar io of open multi-age n t system. Definition 3.1 (T rajecto ry of Points of Interest (TPI)) Consider an open multi-agent system (1) . Assume that for every k ≥ 0 , the equation y = f ( y , V k , V k , E k , E k , u k , u k ) has a unique solution and deno te that solution as x e k . Then, the sequence { x e k } k ∈ Z ≥ 0 is called trajectory of points of interest o f the ope n multi-agent system. Observe that x e k 0 ∈ R V k 0 represents th e hy pothetical equilibriu m o f th e dy namics followed by x k if th e three sequences V k , E k and u k would be kept constant for all k ≥ k 0 . Consequ ently , x e k 0 is determin ed o nly by inf o rmation at time k 0 : the time-variance of V k , E k and u k does not imply any am b iguity in the definition of the seq uence x e k . The next defin ition introduc es a notion akin to a weak fo rm of L yapu nov stability for open m u lti-agent systems. Definition 3.2 (O pen Sta bility o f a T rajecto ry of Points of Interest) Let x k be the evolution o f an open multi-agent system. A trajectory of points o f interest x e ( t ) is said to be op e n stable if ther e exists a stability rad ius R ≥ 0 with the following pr operty: for every ε > R , th e re exis ts δ > 0 suc h tha t if 1 √ n 0 || x 0 − x e 0 || < δ , then 1 √ n k || x k − x e k || < ε for every k ≥ 0 . Note tha t in this defin ition distances are no rmalized by the nu m ber of ag ents. Such normalizatio n , which is tri vial when the set of agents is in variant, is useful here because it allows for a fair comp arison of distances ev aluated in spaces of d ifferent dimen sio n . B. Con tractive OMAS Next we define a par ticular c lass of open -multi-agen t systems of our inte r est. Definition 3 .3 (Cont r active O MAS) Conside r th e open multi-agent system in (1) . The OMAS is said to be contractive if there exists γ ∈ [0 , 1 ) such that for all x, y ∈ R V k and for all k ≥ 0 || f ( x, V k , V k , E k , u k ) − f ( y , V k , V k , E k , u k ) || ≤ γ || x − y || . (5) Novem ber 27, 2024 DRAFT 6 By Banach Fixed Point Theo rem, ev ery contr activ e OMAS has a TPI. As an example, consider system (3). Unde r Assumption 2.1, the OPDC is a c o ntractive OMAS a nd the solution x e k is uniq ue for ev ery k and can b e compu ted as x e k = ( I − P k ) − 1 αu k =  I + ε α L k  − 1 u k . (6) C. Open distance function Next, we define a so-ca lled “open” distance fu nction which is used to ev aluate the distance be twe en two p oints with labeled co mponen ts that belong to Euclidean space s of different dimensions. I n the par ticu lar case in which the two po ints have compo nents with the same labels, i.e. , the same agents, the op e n distan c e func tio n re d uces to the Eu c lid ean d istance. Definition 3. 4 (Open distance function) Let V 1 and V 2 be two fi nite sets of node in dices. Let d : R V 1 × R V 2 → R ≥ 0 be define d as d ( x, y ) = s X v ∈ V 1 ∩ V 2 ( x v − y v ) 2 + X v ∈ V 1 \ V 2 ( x v ) 2 + X v ∈ V 2 \ V 1 ( y v ) 2 (7) for any x ∈ R V 1 and y ∈ R V 2 . V ariants of Defin ition 3 . 4 can be given based on n orms d ifferent from the 2 -norm. The open distance (7) satisfies se veral natural proper ties, which we summarize in the next statement. Proposition 3.5 (Properties o f o pen distance functio ns) Function d ( x, y ) in (7) is such that for any ve ctors x , y , and z of possibly differ ent dimensions: 1) d ( x, y ) ≥ 0 ; 2) d ( x, y ) = d ( y , x ) ; 3) If x = y , then d ( x, y ) = 0 ; 4) d ( x, z ) ≤ d ( x , y ) + d ( y , z ) Pr o of: Properties 1), 2 ), and 3) b eing evident, we now prove pro perty 4) , i.e., the tria n gle inequa lity . Consider sets V x , V y , V z and d efine the u n ion set R = V x S V y S V z and new vectors ¯ x, ¯ y , ¯ z ∈ R R where their generic compo nent is defined as ¯ x v = x v if i ∈ V x and ¯ x v = 0 o therwise. Since ¯ x , ¯ y , ¯ z b e long to the same space R R , it fo llows that the triangle inequality d ( ¯ x, ¯ y ) ≤ d ( ¯ x, ¯ z ) + d ( ¯ z , ¯ y ) holds true since the open distance reduces to the ordinary Euclidean o ne. The result follows because one can readily verify th at d ( ¯ x, ¯ y ) = d ( x, y ) .  Note that the converse of the third implication (iden tity o f indiscernibles) does not hold. Ind eed, co nsider x ∈ R { 1 , 2 } to be x = [1 , 0] an d y ∈ R { 1 } to be [1] . T hen, d ( x, y ) = 0 d espite the two vectors being different. Having th is open distance av ailable, we can naturally u se it on o pen sequen ces to gi ve the f ollowing definition . Novem ber 27, 2024 DRAFT 7 Definition 3 .6 (Open sequence of bo unded varia tion) A sequ ence { y k } of points y k ∈ R V k is said to have bound ed variation if ther e exists a co nstant B ≥ 0 such that d ( y k +1 , y k ) ≤ p | V k +1 | B for all k ∈ Z ≥ 0 . Note that this definition in fact norma lize s the o pen distance by the number o f com ponents o f the vectors. An imp o rtant special case is the tra je c to ry of points of interest of an o pen multi-agent system. Definition 3.7 (T r a jectory of points of interest (TPI) of bounded variation) A TPI { x e k } is said to have bo u nded variation if th er e exists B such tha t d ( x e k +1 , x e k ) ≤ p | V k +1 | B for all k ∈ Z ≥ 0 . D. S tability: sufficient conditions In order to p rovide a sufficient co ndition to ensu re stability in the above sense, we will n eed to co m bine assumptions on both the associated T PI and on its jo in process. The latter assump tio n will take the following form. Definition 3.8 (Bounded jo in process) A join pr ocess is said to be bound ed if e a ch agent joins the OMAS with a state value su ch tha t s X v ∈ A k  x v k +1 − x e,v k +1  2 ≤ p | V k +1 | H ∀ k ∈ Z ≥ 0 for some H ≥ 0 . W e are now r eady to state ou r main stability result. Theorem 3. 9 (Stability of O pen Multi-Ag ent Systems) Con sider an open mu lti-agent system as in (1) with state trajectory { x k } . Assume tha t 1) the OMAS is contractive with pa rameter γ ∈ [0 , 1) ; 2) its TPI { x e k } ha s b ounded variation with con stant B ; 3) the join pr o cess is bou nded with constant H ; 4) | V k +1 | ≥ β 2 | V k | for a ll k with β > γ . Then, the trajectory o f p oints of interests is open stable (Defin ition 3 .2) with stability radius R = B + H 1 − γ β Pr o of: At each iteration k the agents first update their state, then some new agents m ay join and some may Novem ber 27, 2024 DRAFT 8 leav e. By consid e r ing the open d istance fun ction, it holds d ( x k +1 , x e k +1 ) = s X v ∈ V k +1 ∩ V k ( x v k +1 − x e,v k +1 ) 2 + X v ∈ V k +1 \ V k ( x v k +1 − x e,v k +1 ) 2 ≤ s X v ∈ V k +1 ∩ V k ( x v k +1 − x e,v k +1 ) 2 + s X v ∈ V k +1 \ V k ( x v k +1 − x e,v k +1 ) 2 ≤ s X v ∈ V k +1 ∩ V k ( x v k +1 − x e,v k ) 2 + s X v ∈ V k +1 ∩ V k ( x e,v k +1 − x e,v k ) 2 + s X v ∈ V k +1 \ V k ( x v k +1 − x e,v k +1 ) 2 . (8) Since the OMAS is contractive, we ob serve that s X v ∈ V k +1 ∩ V k ( x v k +1 − x e,v k ) 2 ≤ γ s X v ∈ V k +1 ∩ V k ( x v k − x e,v k ) 2 ≤ γ d ( x k , x e k ) . (9) Now , n ote that the tr ajectory of poin ts o f interest is of bounded variation, imply ing s X v ∈ V k +1 ∩ V k ( x e,v k +1 − x e,v k ) 2 ≤ d ( x e k +1 , x e k ) ≤ p | V k +1 | B , (10) and that the join process is bo u nded as per Definition 3.8, imply ing s X v ∈ V k +1 \ V k ( x v k +1 − x e,v k +1 ) ≤ p | V k +1 | H. (11) Thus, by u pper bound ing the rig hthand side of (8) by (9)-(10)-(11), we can write d ( x k +1 , x e k +1 ) ≤ γ d ( x k , x e k ) + p | V k +1 | B + p | V k +1 | H. (12) Let us n ow divide b o th sides of (1 2) b y the squ are root of th e cardinality of | V k +1 | d ( x k +1 , x e k +1 ) p | V k +1 | ≤ γ d ( x k , x e k ) p | V k +1 | + B + H. (13) By assumptio n , | V k +1 | ≥ β 2 | V k | where β > γ , th us we can write d ( x k +1 , x e k +1 ) p | V k +1 | ≤ γ β d ( x k , x e k ) p | V k | + B + H. (14) This ineq uality im plies that theTPI is open stable with stability radius R = B + H 1 − γ β .  Novem ber 27, 2024 DRAFT 9 I V . A P P L I C A T I O N : O P E N P RO P O RT I O NA L D Y NA M I C C O N S E N S U S W e now cha racterize the conv ergence proper ties of the Op en Propor tional Dy namic Consensus proto col. Theorem 4.1 (Stability of Open Proportional Dynamic Consensus) Consider the Open Pr oportional Dyn amic Consensus a lgorithm (OPD C) und er Assumption 2.1 a n d a ssume tha t β > 1 − α . Let ¯ u k = 1 T u k n 1 , ˆ u k = u k − ¯ u k . If the sequ ence of r efer ence signals { u k } satisfies k ˆ u k k ∞ ≤ Π , Π ≥ 0 (15) and d ( ¯ u k +1 , ¯ u k ) ≤ p | V k +1 | U, (16) then the OPDC is open sta b le with stab ility radius R =  1 + 2 1+ ε α ¯ λ 2 + 1 β 1 1+ ε α ¯ λ 2  Π + U 1 − 1 − α β Pr o of: The proo f is divided into fo ur step s which lead to th e applicatio n o f Theorem 3.9. Step 1 : As we have already o bserved right before (6), system (3) u n der Assump tio n 2.1 is a contractive OMAS with γ = 1 − α . Step 2: I f the sequ e nce of r eference signals u k (16) satisfies (15) and ( 16), then th e TPI is of bound ed v ariation with con stant B = 1 1 + ε α ¯ λ 2  1 + 1 β  Π + U. W e start the pr o of of step 2 by exploiting the triangle inequality pr operty o f the op en d istance function d ( x e k +1 , x e k ) ≤ d ( x e k +1 , ¯ u k +1 ) + d ( x e k , ¯ u k ) + d ( ¯ u k +1 , ¯ u k ) . (17) The points of interest are x e k = ( I − P k ) − 1 αu k = ( αI + εL k ) − 1 α ( ¯ u k + ˆ u k ) . (18) Since ( αI + εL k ) − 1 ¯ u k = ¯ u k for any L k we can write x e k − ¯ u k = α ( αI + εL k ) − 1 ˆ u k . (19) Now , sin c e the eigenvector correspo n ding to the largest eigenv alue of ( αI + εL k ) − 1 is 1 and 1 T ˆ u k = 0 , it holds k x e k − ¯ u k k 2 = k α ( αI + εL k ) − 1 ˆ u k k 2 ≤ 1 1 + ελ 2 ,L k α k ˆ u k k 2 , (20) where  1 + ελ 2 ,L k α  − 1 is the seco nd largest eige nvalue of matrix α ( αI + ε L k ) − 1 . Then, the d istance between the po int of interests an d the referen ce sign al at time k satisfies d ( x e k , ¯ u k ) = k x e k − ¯ u k k 2 ≤ α α + ελ 2 ,L k k ˆ u k k 2 ≤ 1 1 + ε α λ 2 ,L k p | V k |k ˆ u k k ∞ . (21) Novem ber 27, 2024 DRAFT 10 By noting th at | V k +1 | β 2 ≥ | V k | , k ˆ u k k ∞ ≤ Π and d ( ¯ u k +1 , ¯ u k ) ≤ p | V k +1 | U f or some U ≥ 0 , it follows from (17) that d ( x e k +1 , x e k ) ≤ p | V k +1 |  1 1 + ε α ¯ λ 2  1 + 1 β  Π + U  = p | V k +1 | B . (22) Step 3: The join process of the OPCD is b ounded accordin g to De fin ition 3.8 with H = 1 + 1 1 + ε ¯ λ 2 α ! Π . In the OPDC algo rithm new ag ents join with a state value equal to th eir re f erence sign al. Sinc e fro m (20) at time k + 1 , k x e k +1 − ¯ u k +1 k 2 ≤ α α + ελ 2 ,L k k ˆ u k +1 k 2 , and | u v k +1 − ¯ u k +1 | ≤ Π b y assumptio n, b y recalling that A k +1 = V k +1 \ V k , it h o lds s X v ∈ A k +1  x v k +1 − x e,v k +1  2 ≤ p | V k +1 \ V k | (1 + α α + ελ 2 ,L k +1 )Π ≤ p | V k +1 | (1 + 1 1 + ε ¯ λ 2 α )Π Thus, the join process of the OPCD algorithm is bou n ded acco rding to Definition 3 . 2 with H = 1 + 1 1 + ε ¯ λ 2 α ! Π . Step 4 : By Th eorem 3 .9, the TPI of th e OPCD algo rithm is ope n stable with stability radius R = B + H 1 − γ β .  When ¯ λ 2 = 0 , that is, the jo in pr ocess do es no t gu a rantee a unifo rm connectivity , then the stability radius in Theorem 4.1 ta kes the simpler form R 0 = (3 β + 1)Π + β U α + β − 1 and we o bserve that in general R ≤ R 0 . V . N U M E R I C A L E X A M P L E S In this section we show a numerical example of the OPCD algorithm. Our simulations are perfor med as f ollows. W e c o nsidered as tu ning parameters ε = 0 . 01 , α = 0 . 1 . The simu lated scenario consists of a network of 2 00 agents at the initial time, w ith initial values cho osen un if ormly at rando m in the interval [ − 5000 , 500 0] . The initial graph is generated as an Erd ˝ os-R ´ enyi graph with ed ge pr obability p = 0 . 0 5 . At each iteration o ne agent leaves with probab ility 0 . 06 an d one agent joins with probability 0 . 1 : each arriving agen t creates r a ndom edges with probability 0 . 05 with all other agents. Input referen ce signals are constant and, when agents join the network, are ch osen unifor m ly at ran dom in the inte r val [0 , 1] . After describing our simu lation setup, we pr esent one typ ical realization. T o begin with , in Fig u re 1 we show the ev olution of the nu mber of agents and in Figu re 2 we sho w the e v olution of the normalized o pen distance d ( x e k , ¯ u 1 ) √ | V k | , Novem ber 27, 2024 DRAFT 11 50 100 150 200 250 Time 198 200 202 204 206 208 210 Number of agents Fig. 1. Time-v ar ying number of agents. that is, the d istan ce between the current poin t of inter e st and the a verage of the inpu t reference signals gi ven to the agents. The value of this quantity depends o n the OPCD parameters, in particu lar it could be r e duced by decrea sin g the par ameter α . W e then p roceed to exemplify the stability pro perties of the OPDC. T o th is pu rpose, Figure 3 shows the evolution of the n ormalized open distance d ( x k ,x e k ) √ | V k | , that is, the distance of th e state x k of the n etwork from the cur rent point of interest x e k . It c an be observed that this n ormalized distance rem a ins bounde d after a transient decrease. This phenom e non is con sistent with the stability analysis gi ven in The orem 4.1. E ven th ough our analysis makes deterministic assumption s and therefore does not in p rinciple allow drawing conclusions on this random ized e v olu- tion, we can a posteriori verify th at the simulated join/leave process h as satisfied the assumptions of T heorem 4.1 with min im um algebr aic connectivity ¯ λ 2 = 0 . 9 037 , | V k +1 | ≥ β 2 | V k | with β = 0 . 9975 , largest d egre e equal to 20 , Π = 0 . 5139 , and U = 0 . 00017 85 .The r efore, the result implies a stability rad ius eq ual to R = 17 . 375 , which appears to be a conservativ e estimate acco rding to Figure 3. Moreover , in Figure 4 we show the ev olution of the nor malized op en distance d ( x k , ¯ u 1 ) √ | V k | , which rep r esents the distance between the network state an d the average o f the input ref erence sign a ls. Estimating the latter q uantity is the objective of the OPCD protocol. T his estimation error can be seen to conv erge to a boun ded value despite the open nature o f the multi- agent system. For a useful comp a rison, in Figures 5 and 6 we show a simulation with of the PDC a lgorithm with a fi xed set of agents ( n = 200 ) and con stant reference signals. In Fig u re 6 it can be seen that the n etwork state con verges to its equilibriu m point (up to m achine precision ) , in con trast with the finite error in Figure 3. In Figu re 5 it can be seen that the network state converges to a steady -state which has a bou nded er ror with respect to the average refe rence signals: in com parison with Figu re 4 , the Open PDC rea c hes a similar steady -state err or ( albeit at slower pace) as its classical PDC counterpart. V I . C O N C L U S I O N S In this paper we pro p osed a theor etical f ramework for stability analysis of discrete-time open multi-agent systems. Standard system-theoretic tolls d o not apply directly to OMAS, beca use o f th e evolution of their state space. For this reason, we had to pro pose sev eral new definitions, includin g suitable definitions of state ev olution and o f stability . The prop osed no tion of stability has two features: (1) the distance from the origin is norma lize d by the number of ag e nts; an d (2 ) the definitio n disregard s what happens within a certain distance from the o rigin ( we refer Novem ber 27, 2024 DRAFT 12 50 100 150 200 250 Time 0.13 0.14 0.15 0.16 0.17 d ( x e k , ¯ u 1 ) √ | V k | Fig. 2. Evolution of the normaliz ed open distance between av era ge re ference input and point of intere st: d ( x e k , ¯ u 1 ) √ | V k | . 50 100 150 200 250 Time 10 -4 10 -2 10 0 10 2 10 4 d ( x k ,x e k ) √ | V k | R Fig. 3. Evolution of the normaliz ed open distance between netw ork s tate and point of interest : d ( x k ,x e k ) √ | V k | . 50 100 150 200 250 Time 10 0 10 1 10 2 10 3 10 4 10 5 d ( x k , ¯ u 1 ) √ | V k | Fig. 4. Evolution of the normaliz ed open distance between netw ork s tate and average reference inp ut: d ( x k , ¯ u 1 )) √ | V k | . 0 50 100 150 200 250 Time 10 0 10 1 10 2 10 3 10 4 10 5 d ( x k , ¯ u 1 ) √ | V k | Fig. 5. Evolut ion of the normal ized ope n distance bet ween network sta te and average reference input: d ( x k , ¯ u 1 )) √ | V k | in the ca se of time-in v aria nt number of age nts n = 200 . to this distanc e as stability rad ius). In order to study the evolution an d the stab ility of OMAS, it is necessary to compare states that belo ng to different spaces. T o this purpo se, we defined the open distance fu nction and u sed it to establish c riteria for stability in the p roposed o pen scenario. I n particular , we showed that m ulti-agent systems whose dyn a mics (up to arriv als and d epartures o f age n ts) can b e defined by contraction maps are stable acco r ding Novem ber 27, 2024 DRAFT 13 0 50 100 150 200 250 300 350 400 450 500 Time 10 -16 10 -14 10 -12 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 d ( x k ,x e k ) √ | V k | Fig. 6. Evolution of th e normalized open distanc e be tween network state and point of interest: d ( x k ,x e k ) √ | V k | in the case of time -in v ariant number of agents n = 200 . to our de fin ition and their stability radius d e pends up on the prop erties of the jo in and leav e mechanisms in th e network. Fur thermore , we ap plied ou r r e sults to an adapta tio n to OMAS o f the pro portion a l dynamic consen su s protoco l. 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