Topology-Dependent Stability of a Network of Dynamical Systems with Communication Delays

Transcription

1 Proceedings of the Europen Control Conference 7 Kos, Greece, July -5, 7 TuC.3 Topology-Dependent Stbility of Network of Dynmicl Systems with Communiction Delys Angel Schöllig, Ulrich Münz, Frnk Allgöwer Abstrct In this pper, we nlyze the stbility of network of first-order liner time-invrint systems with constnt, identicl communiction delys. We investigte the influence of both system prmeters nd network chrcteristics on stbility. In prticulr, non-conservtive stbility bound for the dely is given such tht the network is symptoticlly stble for ny dely smller thn this bound. We show how the network topology chnges the stbility bound. Exemplrily, we use these results to nswer the question if symmetric or skew-symmetric interconnection is preferble for given set of subsystems. Index Terms Network of dynmicl systems, time-dely, network topology, stbility. I. INTRODUCTION The intersection of control nd informtion technology hs become one of the most ctive fields in the control community. Communiction networks offer the possibility to esily connect lrge sets of dynmicl systems. This is prticulrly interesting when solving complex control tsks using decentrlized controllers. Exmples cn be found in mny fields of ppliction, like coopertive control of groups of uninhbited utonomous vehicles UAV, decentrlized control of vst chemicl plnts, or interconnection of systems nd controllers in crs. All these pplictions re incresingly importnt, but their rigorous nlysis nd design come long with lot of problems. They minly rise from the big gp between control nd informtion theory. In the considered re, most of the work on lrge sets of dynmicl systems cn be divided in two ctegories: coopertive control nd decentrlized control. The coopertive control of sets of independent dynmicl systems ims t chieving generl tsks like consensus or formtion building [] []. In these works, locl control lw for ech subsystem is proposed. Then it is shown tht given objective is chieved despite some network deficiencies like pcket delys, pcket loss, or switching topologies. The design of decentrlized controllers for sets of interconnected dynmicl systems with delys goes bck to the 8s. In most cses, it is ssumed tht the interconnection between the subsystems is delyed nd there is no communiction between the controllers [] [3]. Recent works expnd these ides to decentrlized networked controllers [], [5]. Our work dels with decentrlized networked control of lrge sets of dynmicl systems. However, we re not considering the design of locl controllers. Rther, we study the stbility of the obtined network of dynmicl systems A. Schöllig, U. Münz, nd F. Allgöwer re with the Institute for Systems Theory nd Automtic Control, University of Stuttgrt, Germny. {muenz,llgower}@ist.uni-stuttgrt.de U. Münz s work is finncilly supported by The MthWorks Foundtion. NDS. How does the stbility of this NDS depend on the prmeters of the subsystems nd of the network? In fct, this question expresses one of the principl difficulties when combining control nd informtion technology: how do they effect ech other? In this pper, we consider n rbitrry number of identicl subsystems modeled s first-order LTI systems. The network is represented by directed grph nd exhibits constnt nd identicl communiction delys in ll chnnels. Constnt communiction delys re gurnteed by certin communiction protocols. Our min result gives the exct stbility bound τ such tht this NDS is symptoticlly stble for ny dely τ [, τ, i.e., the system is not symptoticlly stble for τ = τ. We derive n nlyticl formul tht reltes τ to the prmeters of the subsystems nd the eigenvlues of the djcency mtrix of the underlying grph. Exemplrily, we use this result to describe the principl difference between symmetric nd skew-symmetric interconnections. The reminder of this pper is structured s follows: In Section II, we present some bsic mteril on the stbility nlysis of time-dely systems nd on lgebric grph theory. The problem sttement is given in Section III. The exct stbility bound τ is derived in Section IV. In Section V, we show how this result cn be used to investigte different network topologies. The pper is concluded in Section VI. II. PRELIMINARIES A. Stbility of Time-Dely Systems A good introduction to time-dely systems TDS is given in [6], [7]. More detils on the discussed topics cn be found there. We consider the following liner time-invrint LTI TDS with dely τ ẋt = A xt+a xt τ, t, xθ = x θ, θ [ τ,], with xt R n, A,A R n n, τ nd initil condition x θ, continuous function mpping [ τ,] to R n. The stbility of this TDS cn be studied efficiently using frequency domin methods. In the sme spirit s for dely-free LTI systems, chrcteristic qusipolynomil ps, e τs of is derived using the Lplce trnsform: ps, e τs =det si A A e τs. Remember tht the stbility exponent α α := mx { Res ps, e τs = } 3 ISBN

2 TuC.3 is continuous with respect to τ, see [6], nd we hve the following result: Lemm [6], Thm..5: System is symptoticlly stble if nd only if α <. In other words, ll roots of must be in the open left hlf plne C <. Note tht hs infinitely mny roots. For simplicity, we cll system stble if it is symptoticlly stble nd otherwise unstble, s done for exmple in [6]. We ssume tht system is stble without dely, i.e., ll roots of re in C < for τ =.Weincreseτ until the system becomes unstble, i.e. α =, nd define the stbility bound τ := min { τ pj,e jτ =for some R }. Obviously, the systems is stble for ny τ [, τ nd becomes unstble for τ = τ. Note however tht this does not imply tht the system is unstble for ll τ [τ,. Whenever τ <, the system is clled dely-dependent stble. Itissidtobedely-independent stble, ifis stble for ll τ. The pirs k,τ k stisfying the eqution pj k,e j kτ k = 5 hve some importnt properties discussed in [6] nd [8]: There is only finite number of possible zero-crossing frequencies k. Since is rel qusipolynomil, ll its complex roots pper in complex conjugte pirs. Consequently, it suffices to consider only k >. Note tht the frequency =is not possible zero-crossing frequency becuse we ssume tht the system is stble for τ =, i.e., pj, p,. There re mny different wys to compute τ.we hve studied vrious of them nd the most suitble for the considered problem is the frequency-sweeping test, e.g. [6]. The nme frequency-sweeping test refers to stbility nlysis using the eigenvlues of frequency-dependent mtrix. Theorem [6], Thm..: System is stble independent of dely if nd only if i A is stble, ii A + A is stble, nd iii ρ j I A A <, >, where ρ denotes the spectrl rdius of mtrix. Theorem [6], Thm..: Suppose tht system is stble t τ =nd tht the mtrix j I A is invertible for ll,. Furthermore, define min τ ki if λ k j ki I A A = e jθ ki i=,...,n for some ki,, θ ki [, π] τ k := if ρ j I A A <,,, 6 where θ ki = ki τ ki.then,τ =min k=,...,n τ k, i.e., system is stble for ll τ [, τ but becomes unstble t τ = τ. B. Bsic Algebric Grph Theory A directed grph G =V, E of order n is defined by set of vertices V = {v,v,..., v n }, nd set of edges E V V. The edges of G re denoted by e ij =v i,v j E, i.e., there is n edge from vertex j to vertex i. We consider directed grphs without loops, i.e., e ii / E for ll i I= {,,...,n}. Moreover, we define the djcency mtrix C = [c ij ] R n n. Ech djcency element c ij is ssocited with n edge e ij, i,j I. We require for ll edges e ij E c ij. More detils on lgebric grph theory cn be found for exmple in [9]. Among others, we consider the well-known ring, str, nd complete topology, shown in Tble I for the cse n =.The eigenvlues of these topologies re given by cf. [] [λc ring ] n =, λ, C str =± n, λ 3,..., n C str =, λ C compl =n, λ,..., n C compl =, where C ring,c str,c compl R n n. III. PROBLEM STATEMENT A NDS consisting of n dynmicl subsystems cn be represented by S =Σ,G,T. The behvior of the dynmicl subsystems is described by Σ = Σ,...,Σ n, e.g., using differentil equtions. The input nd output of ech subsystem re u i nd y i, respectively. The topology of the network is given by the grph G of order n, wherethevertex v i of G corresponds to subsystem Σ i. The communiction dely is represented by T = [τ ij ] R n n such tht c ij = τ ij =for ll i, j I. In the most generl cse, itmybessumedthtτ ij is time-vrint or stochstic. A communiction chnnel from Σ j to Σ i is modeled by n edge e ij Ewith mplifiction c ij nd communiction dely τ ij. In other words, the input u i t of system Σ i is function of the weighted outputs c ij y j t τ ij of the neighboring systems Σ j with e ij E. The stbiliztion of such NDS S hs for exmple been ddressed in [5], yet without delys. In this pper, we ssume tht the dynmicl representtion of ll subsystems Σ i,i I, is the sme nd cn be expressed s ẋ i t =x i t+bu i t Σ i : y i t =x i t 7 x i = x i, with, b, x i,u i R. Our min results, Theorems 3 nd, cn be expnded to heterogeneous higher order LTI systems. Yet, the connection between the dynmicl behvior of the subsystems nd the network topology is much clerer for this first-order model. Moreover, since ll subsystems behve in the sme wy, we cn derive stbility conditions for rbitrrily lrge networks depending only on, b, nd n. Furthermore, we ssume tht the input u i t of ech subsystem depends linerly on the mplified outputs nd tht 98

3 TuC.3 TABLE I TYPICAL NETWORK STRUCTURES:THE RING, STAR, AND COMPLETE TOPOLOGY FOR VERTICES. C ring = B A C str = B A C compl = B A ll delys re constnt nd identicl, τ ij = τ, i,j I,i.e. n u i t = c ij y j t τ, i I. 8 j= Constnt nd identicl delys re obtined for exmple using communiction protocols with gurnteed mximul dely. This ssumption llows us to determine non-conservtive nlyticl stbility bounds τ. This is in generl not possible for multiple heterogeneous delys becuse clculting the exct stbility bounds of TDS with incommensurte delys is n NP-hrd problem [6]. The networked system 7 with inputs 8 cn be written in the following compct mtrix representtion ẋt =xt+bcxt τ, 9 where x R n, C R n n nd, b, τ R, τ. Notice tht 9 describes the dynmics of the NDS completely. It contins the informtion bout the dynmicl subsystems,, b, their quntity n, the network topology C, nd the communiction dely τ. With the frequency-sweeping test, we cn now investigte the stbility of 9 subject to the prmeters, b, n, nd τ s well s the eigenvlues of C. IV. MAIN RESULT In this section, we derive the stbility bound τ for our NDS 9 bsed on the frequency-sweeping test introduced in Section II-A. We study the cse of dely-independent nd dely-dependent stbility seprtely in Sections IV-A nd IV- B, respectively. Therefore, we define the dely-independent stbility set S DI C nd the dely-dependent stbility set S DD C s follows S DI C :={, b 9 is dely-independent stble}. S DD C :={, b τ, such tht 9 is stble τ [, τ but unstble for τ = τ}. Note tht S DI S DD is the set where 9 is stble for τ =. A. Dely-Independent Stbility The dely-independent stbility condition is direct result of Theorem with A = I nd A = bc. Theorem 3: NDS 9 is stble independent of dely if nd only if i + b Re [λ k C] <, k I, nd ii b ρ C. Proof: The first condition of Theorem requires <. The second condition trnsfers to Re [λ k I+ bc] < + b Re [λ k C] <, for ll k I. For the lst condition, we hve b ρ C < +, > b ρ C. Note tht < is necessry for both nd to be true. B. Dely-Dependent Stbility The dely-dependent stbility set is given in the following theorem. Theorem : There exists τ, such tht 9 is stble for ll τ [, τ nd becomes unstble for τ = τ if nd only if i + b Re [λ k C] <, k I, nd ii b ρ C >. Then, τ = τ, b, C is given by k Im λ k C Re λ k C τ =min rccos k k b λ k C subject to b λ k C >,where k = b λ k C. 3 The proof is presented in the ppendix. Remrk : Remember tht Theorem 3 nd re only vlid if the delys in ll chnnels re identicl, i.e., τ ij = τ for ll i, j I. In prticulr, they re not vlid for heterogeneous, i.e. incommensurte, delys τ ij τ. C. Interprettion Note tht both Theorem 3 nd depend only on the eigenvlues of the djcency mtrix C, but not on the mtrix itself. Moreover, Eqution pplies for n rbitrry number n of subsystems. This is remrkble becuse the eigenvlues of the djcency mtrices of typicl network structures cn be expressed s functions of n, see Section II- B. Using these results, we cn esily compre the stbility sets of different topologies with rbitrrily mny subsystems. In ddition, we cn determine how the stbility sets of these topologies chnge with n incresing number of subsystems. We suppose tht n djcency mtrix C is given in order to illustrte the result of Theorem 3. From Conditions i nd 99

4 TuC.3 ii in Theorem 3, it results <. Furthermore, the lines b = ρc nd b = ρc bound the stbility set S DI C. The points, b on these boundries re stble except for the cse tht the eigenvlue λ C with the highest modulus is rel. In this cse, ll points on the upper boundry b = ρc re not in S DI if ρc =λ C. All points on the lower boundry b = ρc re not in S DI if ρc = λ C. If there re two rel eigenvlues λ C nd λ C = λ C with highest modulus, then ll points on both boundries re not in S DI. Note tht the stbility set S DI hs the sme tringulr shpe for ny topology nd depends only on the spectrl rdius ρc. The set gets smller s ρc increses. Two exmples of S DI re depicted in the third column of Tble II. The dely-independent stbility set is drk blue. Next, we consider the ring, str, nd complete topology, presented in Section II-B. By incresing the number of subsystems, the stbility set S DI C ring does not chnge, wheres S DI C str nd S DI C compl become smller. Thereby, the decrese in S DI C compl is lrger thn the decrese in S DI C str. Similr results re obtined for S DD. V. SYMMETRIC AND SKEW-SYMMETRIC ADJACENCY MATRICES In this section, we exploit our min result to compre the stbility of NDS with different interconnections. As n exmple, we consider symmetric nd skew-symmetric djcency mtrices C. After investigting the differences of these mtrices theoreticlly, we explicitly clculte the stbility sets nd the stbility bound τ for two networked systems consisting of three subsystems. The eigenvlues of both symmetric mtrices C s = Cs T nd skew-symmetric mtrices C ss = Css T hve specific properties. Lemm [], Cor..5.: Rel symmetric mtrices C s hve only rel eigenvlues, i.e., Im λ k C s = for ll k I. Rel skew-symmetric mtrices C ss hve only purely imginry eigenvlues, i.e., Re λ k C ss = for ll k I. These properties result in different stbility sets S DD nd S DI even if ll elements of C s nd C ss re the sme except for the sign. This is shown in Exmple. First, we show how the clcultion of τ cn be simplified. Corollry : Suppose NDS 9 with symmetric djcency mtrix C s tht fulfills Conditions i nd ii in Theorem. If C s hs positive nd negtive eigenvlues nd the mximl nd miniml eigenvlue of C s fulfill λ mx C s λ min C s,thens DD C s nd the stbility bound is τ = b λ mc s rccos, 5 bλ m C s with { } λ m C s = λ : λ =mx λ k C s, k I. k Corollry : Suppose NDS 9 with skew-symmetric djcency mtrix C ss tht fulfils Conditions i nd ii in Theorem. Then the stbility bound τ is τ = b ρ C ss rccos b ρ C ss. 6 Corollry nd result from Theorem. The detiled proofs nd more chrcteristics of the stbility sets for symmetric nd skew-symmetric djcency mtrices re presented in []. Note tht the stbility bound τ is the sme for positive nd negtive b in the skew-symmetric cse. In the following, we use these corollries nd the results of Section IV to nlyze exemplrily two NDS with symmetric nd skew-symmetric djcency mtrices. Both of them hve complete topology. Exmple : Consider the djcency mtrices C s = nd C ss =, 7 which represent NDS consisting of three subsystems. The network topologies of C s nd C ss re depicted in the second column of Tble II. We use solid line to show tht c ij = nd dshed line for edges with c ij =. The eigenvlues of the mtrices 7 re nd λ, C s =, λ 3 C s = 8 λ, C ss =±j 3, λ 3 C ss =. 9 The stbility sets S DI drk blue nd S DD grey of both systems re shown in the third column of Tble II. Note tht there is smll difference in S DI ndbigdifference in S DD. Both systems re unstble for >. Inthelst column of Tble II, we present the stbility bounds τ for both systems. In these grphics, ll vlues τ 8 re set to τ =8. Remember tht in the skew-symmetric cse τ is the sme for positive nd negtive b. Therefore, we only present b. Note tht the stbility bound τ is higher for the symmetric djcency mtrix s long s, b remins inside S DD C s. In conclusion, we re ble to efficiently nlyze the stbility of NDS pplying the conditions of Section IV. Moreover, we re ble to design stble NDS: For given prmeters, b, we cn determine which topology provides the best stbility properties. For Exmple, if we suppose = nd b =.75, then we hve τ.6 for the symmetric cse nd τ.6 for the skew-symmetric cse. Hence, the symmetric topology is preferble. On the contrry, if we hve = nd b =.5, then the skewsymmetric topology gurntees stbility up to τ =.65 nd the symmetric topology is unstble even for τ =. Another simple design tsk is to select n djcency mtrix nd input gin b for given nd τ. Of course, we could choose C nd b such tht the networked system lies in S DI. However, we might get better performnce in terms of settling time if we choose strongly negtive b in S DD.For

5 TuC.3 TABLE II STABILITY OF NDS 9 COMPARISON OF A SYMMETRIC VS. ASKEW-SYMMETRIC ADJACENCY MATRIX OF A NETWORK WITH 3 NODES:ADJACENCY MATRIX FIRST COLUMN, GRAPH TOPOLOGY SECOND COLUMN, STABILITY SETS S DI DARK BLUE AND S DD GREY IN THE THIRD COLUMN, AND PLOTS OF THE STABILITY BOUND τ ON THE PARAMETER SET, b [, ] [, ] LAST COLUMN. C s = b τ b C ss = b τ b Exmple, we ssume = nd τ =nd get b =.76 for the skew-symmetric djcency mtrix nd b =.99 for the symmetric cse. Hence, C s is the better choice. However, if we hve = nd τ =., thenwegetb =.88 for C ss nd gin b =.99 for C s remember tht this is the bound of S DD C s. Obviously, C ss is better suited. VI. CONCLUSIONS In this pper, we nlyzed the stbility of network of dynmicl systems. In prticulr, we consider n rbitrry number of identicl first-order LTI systems tht re interconnected vi network with identicl communiction delys. Our min result gives the exct stbility bound τ such tht the NDS is symptoticlly stble for ny dely τ [, τ but loses this property for τ = τ. The stbility bound is derived nlyticlly subject to the prmeters of the subsystems nd the eigenvlues of the djcency mtrix C of the network. For typicl network topologies, these eigenvlues hve been determined s function of n. Finlly, we studied the principl difference of symmetric nd skewsymmetric djcency mtrices in terms of stbility. Our results pply for generl network of first-order LTI systems nd determine its stbility bound. An expnsion to NDS tht consist of non-identicl higher-order LTI systems is esily possible t the expense of the compct formultion of the result. APPENDIX PROOF OF THEOREM First, we proof the existence of τ. Condition i is obvious becuse 9 must be stble for τ =. Condition ii gurntees tht τ< cf. Theorems nd 3. Next, we show how is derived from Theorem. Note tht j I A is invertible for ll, becuse we hve A = I, R. We first clculte the zero-crossing frequencies ki using the eqution λ k j ki I A A =. Remember tht there is only finite number of zero-crossing frequencies ki Prt II-A. We rewrite using 9 nd obtin ki = k = b λ k C, k I. The rdicnd is positive for t lest one k Ibecuse of Condition ii. Hence, we obtin t lest one rel k >. We clculte τ k for ech rel k using 6 nd obtin ] k τ k = rg [λ k j k bc = rg [ b j k b]+rg [λ k C], 3 }{{} [, π] where rgz denotes the rgument of complex number z. Next, we derive using Condition i, some trigonometric identities, nd the fct tht ll complex eigenvlues of C pper in complex conjugte pirs. For simplicity of nottion, λ k C is replced by λ, k by, nd τ k by τ. The clcultion of the rguments in 3 needs cse differentition. For rg [ b jb], the cses shown in Tble III must be considered. The rgument rg [ b j k b] is zero for b =. However, this cse is not of interest becuse b =mens tht there is no interconnection between the subsystems. Results for =cn be determined by clculting the limit in the cses Tble III. For rg [λ], we hve to differentite the cses presented in Tble IV. Results for Reλ = cn be determined by clculting the limit Reλ in the cses A D Tble IV.

6 TuC.3 TABLE III CASE DIFFERENTIATION FOR THE PARAMETERS, b IN THE CALCULATION OF 3 Cse Prmeter rg [ b jb] <,b< rctn + π <,b> rctn 3 >,b< rctn >,b> rctn + π TABLE IV CASE DIFFERENTIATION FOR THE EIGENVALUES λ k C IN THE CALCULATION OF 3 Cse Prmeter rg [λ] A Reλ <, Imλ rctn B Reλ <, Imλ > rctn C Reλ >, Imλ rctn D Reλ >, Imλ > rctn Imλ Reλ Imλ Reλ Imλ Reλ Imλ Reλ + π + π Exemplrily, we consider the combintion cse 3-D. Eqution gives = b λ nd we obtin Imλ τ = rctn +rctn }{{ } Reλ }{{} > > Re λ = rccos + rccos b λ λ Im λ Re λ = rccos b λ. For other combintions, the procedure is the sme. Becuse of Condition i nd the fct tht the eigenvlues of C hve complex conjugte pirs, some combintions do not hve to be considered. For exmple, the inequlity Reλ > must hold in cse 3 Condition i. All possible combintions result in Eqution. The time-dely τ is the minimum of ll clculted τ k. [6] R. Olfti-Sber nd R. M. Murry, Consensus problem in networks of gents with switching topology nd time-delys, IEEE Trnsctions on Automtic Control, vol. 9, pp ,. [7] J. A. Mrshll, M. E. Broucke, nd B. A. Frncis, Formtions of vehicles in cyclic pursuit, IEEE Trnsctions on Automtic Control, vol. 9, pp ,. [8] A. Jdbbie, J. Lin, nd S. Morse, Coordintion of groups of mobile utonomous gents using nerest neighbor rules, IEEE Trnsctions on Automtic Control, vol. 8, pp. 988, 3. [9] H. G. Tnner, A. Jdbbie, nd J. Ppps, Stble flocking of mobile gents, prt I: Fixed topology, in Proceedings of the th IEEE Conference on Decision nd Control, Mui, USA, 3, pp. 5. [], Stble flocking of mobile gents, prt II: Dynmic topology, in Proceedings of the th IEEE Conference on Decision nd Control, Mui, USA, 3, pp. 5. [] T. N. Lee nd U. L. Rdovic, Decentrlized stbiliztion of liner continuous nd discrete-time systems with delys in interconnections, IEEE Trnsctions on Automtic Control, vol. 33, pp , 988. [] J. H. Prk, Robust nonfrgile decentrlized controller design for uncertin lrge-scle interconnected systems with time-delys, Journl of Dynmic Systems, Mesurement, nd Control, vol., pp ,. [3] H. Wu, Decentrlized dptive robust control for clss of lrge-scle systems including delyed stte perturbtions in the interconnections, IEEE Trnsctions on Automtic Control, vol. 7, pp ,. [] M. Rotkowitz nd S. Lll, A chrcteriztion of convex problems in decentrlized control, IEEE Trnsctions on Automtic Control, vol. 5, pp. 7 86, 6. [5] A. Gttmi nd R. Murry, A frequency domin condition for stbility of interconnected MIMO systems, in Proceedings of the Americn Control Conference, Boston, USA,. [6] K. Gu, V. L. Khritonov, nd J. Chen, Stbility of Time-Dely Systems. Boston: Birkhäuser, 3. [7] S.-I. Niculescu, Dely Effects on Stbility: A Robust Control Approch. London: Springer,. [8] N. Olgc nd R. Siphi, An exct method for the stbility nlysis of time-delyed liner time-invrint LTI systems, IEEE Trnsctions on Automtic Control, vol. 7, no. 5, pp ,. [9] C. Godsil nd G. Royle, Algebric Grph Theory. New York: Springer,. [] A. Schöllig, Stbilitätsnlyse von vernetzten Systemen mit Verzögerungen in der Dtenübertrgung, July 6, study thesis, Institute for Systems Theory nd Automtic Control, University of Stuttgrt, Germny. [] R. A. Horn nd C. R. Johnson, Mtrix Anlysis, st ed. New York: Cmbridge University Press, 985. REFERENCES [] R. Olfti-Sber, Flocking for multi-gent dynmic systems: Algorithms nd theory, IEEE Trnsctions on Automtic Control, vol. 5, pp., 6. [] L. Moreu, Stbility of multigent systems with time-dependent communiction links, IEEE Trnsctions on Automtic Control, vol. 5, pp. 69 8, 5. [3] A. Ppchristodoulou nd A. Jdbbie, Synchroniztion in oscilltor networks: Switching topologies nd non-homogeneous delys, in Proceedings of the th IEEE CDC nd ECC, Seville, Spin, 5, pp [] P.-A. Blimn nd G. Ferrri-Trecte, Averge consensus problems in networks of gents with delyed communictions, in Proceedings of the th IEEE CDC nd ECC, Seville, Spin, 5, pp [5] T. Keviczky, F. Borrelli, nd G. J. Bls, Stbility nlysis of decentrlized RHC for decoupled systems, in Proceedings of the th IEEE CDC nd ECC, Seville, Spin, 5, pp