On Event-Triggered Adaptive Architectures for Decentralized and Distributed Control of Large- Scale Modular Systems

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1 Unversty of South Florda Scholar Commons Mechancal Engneerng Faculty Publcatons Mechancal Engneerng 26 On Event-Trggered Adaptve Archtectures for Decentralzed and Dstrbuted Control of Large- Scale Modular Systems Al Albattat Mssour Unversty of Scence and Technology Benjamn C. Gruenwald Unversty of South Florda Tansel Yucelen Unversty of South Florda, Follow ths and addtonal works at: Part of the Mechancal Engneerng Commons Scholar Commons Ctaton Albattat, Al; Gruenwald, Benjamn C.; and Yucelen, Tansel, "On Event-Trggered Adaptve Archtectures for Decentralzed and Dstrbuted Control of Large-Scale Modular Systems" 26). Mechancal Engneerng Faculty Publcatons Ths Artcle s brought to you for free and open access by the Mechancal Engneerng at Scholar Commons. It has been accepted for ncluson n Mechancal Engneerng Faculty Publcatons by an authorzed admnstrator of Scholar Commons. For more nformaton, please contact scholarcommons@usf.edu.

2 sensors Artcle On Event-Trggered Adaptve Archtectures for Decentralzed and Dstrbuted Control of Large-Scale Modular Systems Al Albattat, Benjamn C. Gruenwald 2 and Tansel Yucelen 2, * Department of Mechancal and Aerospace Engneerng, Mssour Unversty of Scence and Technology, Rolla, MO 6549, USA; ata365@mst.edu 2 Laboratory for Autonomy, Control, Informaton, and Systems LACIS), Department of Mechancal Engneerng, Unversty of South Florda, Tampa, FL 3362, USA; bcgruenwald@mal.usf.edu * Correspondence: yucelen@lacs.team; Tel.: Academc Edtor: Dan Zhang Receved: 24 June 26; Accepted: 5 August 26; Publshed: 6 August 26 Abstract: The last decade has wtnessed an ncreased nterest n physcal systems controlled over wreless networks networked control systems). These systems allow the computaton of control sgnals va processors that are not attached to the physcal systems, and the feedback loops are closed over wreless networks. The contrbuton of ths paper s to desgn and analyze event-trggered decentralzed and dstrbuted adaptve control archtectures for uncertan networked large-scale modular systems; that s, systems consst of physcally-nterconnected modules controlled over wreless networks. Specfcally, the proposed adaptve archtectures guarantee overall system stablty whle reducng wreless network utlzaton and achevng a gven system performance n the presence of system uncertantes that can result from modelng and degraded modes of operaton of the modules and ther nterconnectons between each other. In addton to the theoretcal fndngs ncludng rgorous system stablty and the boundedness analyss of the closed-loop dynamcal system, as well as the characterzaton of the effect of user-defned event-trggerng thresholds and the desgn parameters of the proposed adaptve archtectures on the overall system performance, an llustratve numercal example s further provded to demonstrate the effcacy of the proposed decentralzed and dstrbuted control approaches. Keywords: large-scale modular systems; networked control systems; uncertan dynamcal systems; event-trggered control; decentralzed control; dstrbuted control; system stablty and performance. Introducton The desgn and mplementaton of decentralzed and dstrbuted archtectures for controllng complex, large-scale systems s a nontrval control engneerng task nvolvng the consderaton of components nteractng wth the physcal processes to be controlled. In partcular, large-scale systems are characterzed by a large number of hghly coupled components exchangng matter, energy or nformaton and have become ubqutous gven the recent advances n embedded sensor and computaton technologes. Examples of such systems nclude, but are not lmted to, mult-vehcle systems, communcaton systems, power systems, process control systems and water systems see, for example, 6 and the references theren). Ths paper concentrates on an mportant class of large-scale systems; namely, large-scale modular systems that consst of physcally-nterconnected and generally heterogeneous modules. Sensors 26, 6, 297; do:.339/s

3 Sensors 26, 6, of 3.. Motvaton and Lterature Revew Two sweepng generalzatons can be made about large-scale modular systems. The frst s that ther complex structure and large-scale nature yeld to naccurate mathematcal module models, snce t s a challenge to precsely model each module of a large-scale system and the nterconnectons between these modules. As a consequence, the dscrepances between the modules and ther mathematcal models, that s system uncertantes, result n the degradaton of overall system stablty and the performance of the large-scale modular systems. To ths end, adaptve control methodologes 7 3 offer an mportant capablty for ths class of dynamcal systems to learn and suppress the effect of system uncertantes resultng from modelng and degraded modes of operaton, and hence, they offer system stablty and desrable closed-loop system performance n the presence of system uncertantes wthout excessvely relyng on mathematcal models. The second generalzaton about large-scale modular systems s that these systems are often controlled over wreless networks, and hence, the communcaton costs between the modules and ther remote processors ncrease proportonally wth the ncrease n the number of modules and often the nterconnecton between these modules. To ths end, event-trggered control methodologes 4 6 offer new control executon paradgms that relax the fxed perodc demand of computatonal resources and allow for the aperodc exchange of sensor and actuator nformaton wth the remote processor to reduce overall communcaton cost over a wreless network. Note that adaptve control methodologes and event-trggered control methodologes are often studed separately n the lterature, where t s of practcal mportance to theoretcally ntegrate these two approaches to guarantee system stablty and the desrable closed-loop system performance of uncertan large-scale modular systems wth reduced communcaton costs over wreless networks, whch s the man focus of ths paper. More specfcally, the authors of 6,7 23 proposed decentralzed and dstrbuted adaptve control archtectures for large-scale systems; however, these approaches do not make any attempts to reduce the overall communcaton cost over wreless networks usng, for example, event-trggered control methodologes. In addton, the authors of 24 3 present decentralzed and dstrbuted control archtectures wth event trggerng; however, these approaches do not consder adaptve control archtectures and assume perfect models of the processes to be controlled; hence, they are not practcal for large-scale modular systems wth sgnfcant system uncertantes. Only the authors of 3 36 present event-trggered adaptve control approaches for uncertan dynamcal systems. In partcular, the authors of 3,32 consder data transmsson from a physcal system to the controller, but not vce versa, whle developng ther adaptve control approaches to deal wth system uncertantes. On the other hand, the adaptve control archtectures of the authors n consder two-way data transmsson over wreless networks; that s, from a physcal system to the controller and from the controller to ths physcal system. However, none of these approaches can be drectly appled to large-scale modular systems. Ths s due to the fact that large-scale modular systems requre decentralzed and dstrbuted archtectures, and drect applcaton of the results n 3 36 to ths class of systems can result n centralzed archtectures, whch s not practcally desred due to the large-scale nature of modular systems. To summarze, there do not exst reslent adaptve control archtectures for large-scale systems n the lterature to deal wth system uncertantes whle reducng the communcaton costs between the models and ther remote processors..2. Contrbuton The contrbuton of ths paper s to desgn and analyze event-trggered decentralzed and dstrbuted adaptve control archtectures for uncertan large-scale systems controlled over wreless networks. Specfcally, the proposed decentralzed and dstrbuted adaptve archtectures of ths paper guarantee overall system stablty whle reducng wreless network utlzaton and achevng a gven system performance n the presence of system uncertantes that can result from modelng and degraded modes of operaton of the modules and ther nterconnectons between each other. From a theoretcal vewpont, the proposed event-trggered adaptve archtectures here can be vewed

4 Sensors 26, 6, of 3 as a sgnfcant generalzaton of our pror work documented n 35,36 to large-scale modular systems, whch consder a state emulator-based adaptve control methodology wth robustness aganst hgh-frequency oscllatons n the controller response,3, In ths generalzaton, we also adopt necessary tools and methods from 6,23 on decentralzed and dstrbuted adaptve controller constructon for large-scale modular systems. In addton to the theoretcal fndngs ncludng rgorous system stablty and boundedness analyss of the closed-loop dynamcal system and the characterzaton of the effect of user-defned event-trggerng thresholds, as well as the desgn parameters of the proposed adaptve archtectures on the overall system performance, an llustratve numercal example s further provded to demonstrate the effcacy of the proposed decentralzed and dstrbuted control approaches..3. Organzaton The contents of the paper are as follows. In Secton 2, we consder an event-trggered decentralzed adaptve control approach for large-scale modular systems, where the consdered approach assumes that physcally-nterconnected modules cannot communcate wth each other for exchangng ther state nformaton. Specfcally, Theorem and Corollares 4 show the man results of Secton 2 subject to some structural condtons on the parameters of the large-scale modular systems and the proposed event-trggered decentralzed control archtecture see Assumptons 4 and 5). In Secton 3, we consder an event-trggered dstrbuted adaptve control approach n Theorem 2 and Corollares 5 7 for gettng rd of such structural condtons, where the consdered approach assumes that physcally-nterconnected modules can locally communcate wth each other for exchangng ther state nformaton. Fnally, the llustratve numercal example s presented n Secton 4, and conclusons are summarzed n Secton Notaton The notaton used n ths paper s farly standard. Specfcally, R denotes the set of real numbers; R n denotes the set of n real column vectors; R n m denotes the set of n m real matrces; R + denotes the set of postve real numbers; R n n + denotes the set of n n postve-defnte real matrces; S n n denotes the set of n n symmetrc real matrces; D n n denotes the set of n n real matrces wth dagonal scalar entres; ) T denotes transpose; ) denotes nverse; tr ) denotes the trace operator; daga) denotes the dagonal matrx wth the vector a on ts dagonal; and denotes equalty by defnton. In addton, we wrte λ mn A) respectvely, λ max A)) for the mnmum and respectvely maxmum egenvalue of the Hermtan matrx A, for the Eucldean norm and F for the Frobenus matrx norm. Furthermore, we use for the or logc operator and ) for the not logc operator. We adopt graphs 43 to encode physcal nteractons and communcatons between modules. In partcular, an undrected graph G s defned by V G = {,, N} of nodes and a set E G V G V G of edges. If, j) E G, then the nodes and j are neghbors, and the neghborng relaton s ndcated wth j. The degree of a node s gven by the number of ts neghbors, where d denotes the degree of node. Lastly, the adjacency matrx of a graph G, AG) R N N, s gven by: AG) j 2. Event-Trggered Decentralzed Adaptve Control {, f, j) E G, otherwse In ths secton, we ntroduce an event-trggered decentralzed adaptve control archtecture, where t s assumed that physcally-nterconnected modules cannot communcate wth each other. For organzatonal purposes, ths secton s broken up nto two subsectons. Specfcally, we frst brefly overvew a standard decentralzed adaptve control archtecture wthout event-trggerng and then present the proposed event-trggered decentralzed adaptve control approach, whch ncludes rgorous )

5 Sensors 26, 6, of 3 stablty and performance analyses wth no Zeno behavor and generalzatons to the state emulator case for suppressng the effect of possble hgh-frequency oscllatons n the controller response. 2.. Overvew of a Standard Decentralzed Adaptve Control Archtecture wthout Event-Trggerng Consder an uncertan large-scale modular system S consstng of N nterconnected modules S, V G, gven by: S : ẋ t) = A x t) + B Λ u t) + x t)) + δ j x j t)), x ) = x 2) where x t) R n s the state of S, u t) R m s the control nput appled to S, A R n n, B t) R n m are known matrces and the par A, B ) s controllable. In addton, Λ R m m + D m m s an unknown module control effectveness matrx; : R n R m represents matched module bounded uncertantes; and δ j : R n j R m represents matched unknown physcal nterconnectons wth respect to module j, j V G, such that, j) E G. Assumpton. The unknown module uncertanty s parameterzed as: x t)) = W T o β x t)), x R n 3) where W o R g m s an unknown weght matrx, whch satsfes W o F ω, ω R +, and β x t)) : R n R g s a known Lpschtz contnuous bass functon vector satsfyng: wth L β R +. Assumpton 2. The functon δ j x j t)) n Equaton 2) satsfes: β x ) β x 2 ) L β x x 2 4) δ j x j t)) α j x j t), α j >, x j R n j 5) Next, consder the reference model S r capturng a desred closed-loop performance for module, V G gven by: S r : ẋ r t) = A r x r t) + B r c t), x r ) = x r 6) where x r t) R n s the reference state vector of S r, c t) R m s a gven bounded command of S r, A r R n n s the reference system matrx and B r R n m s the command nput matrx. Assumpton 3. There exst K R m n and K 2 R m m, such that A r = A B K and B r = B K 2 hold wth A r beng Hurwtz. Usng Assumptons and 3, Equaton 2) can be equvalently wrtten as: where W ẋ t) = A r x t) + B r c t) + B Λ u t) + W T σ x t), c t)) Λ W T o, Λ + B δ j x j t)) 7) K T, Λ K T 2 T R g +n +m ) m s the unknown weght matrx and σ x t), c t) ) β T x t)), x T t), ct t) T R g +n +m. Motvated from the structure of the uncertan terms appearng n Equaton 7), let the decentralzed adaptve feedback controller of S, V G, be gven by: C : u t) Ŵ t) T σ x t), c t)) 8)

6 Sensors 26, 6, of 3 where Ŵ t) s an estmate of W satsfyng the update law: Ŵ t) γ Proj m Ŵ t), σ x t), c t)) x t) x r t)) T P B, Ŵ ) = Ŵ 9) where Proj m denotes the projecton operator defned for matrces,35,44,45, γ R + beng the learnng rate and P R n n + S n n beng a soluton of the Lyapunov equaton: wth R R n n + S n n. Now, lettng: = A T r P + P A r + R ) e t) x t) x r t) ) W t) Ŵ t) W 2) and usng Equatons 6) and 7), the module-level closed-loop error dynamcs are gven by: ė t) = A r e t) B Λ W Tt)σ x t), c t)) + B δ j x j t)), e t) = e 3) 2.2. Proposed Event-Trggered Decentralzed Adaptve Control Archtecture We now present the proposed event-trggered decentralzed adaptve control archtecture for large-scale modular systems, whch reduces wreless network utlzaton and allows a desrable command trackng performance durng the two-way data exchange between the module S, V G, and ts local controller C, over a wreless network. For ths objectve, we utlze event-trggerng control theory to schedule the data exchange dependent on errors exceedng user-defned thresholds. Specfcally, the module sends ts state sgnal to ts local adaptve controller only when a predefned event occurs. The k -th tme nstants of the state transmsson of the module are represented by the monotonc sequence { } s k k =, where s k R +. The local controller uses ths trggered module state sgnal to compute the control sgnal usng adaptve control archtecture. In addton, the local controller sends the updated feedback control nput to the module only when another predefned event occurs. The j -th tme nstants of the feedback control transmsson are then represented by the monotonc sequence { } r j j =, where r j R +. As depcted n Fgure, each module state sgnal and ts local control nput are held by a zero-order-hold operator ZOH) untl the next trggerng event for the correspondng sgnal takes place. The delay n samplng, data transmsson and computaton s not consdered n ths paper. Consder the uncertan dynamcal module gven by: S : ẋ t) = A x t) + B Λ u s t) + x t)) + δ j x j t)), x ) = x 4) where u s t) R m s the sampled control nput vector. Usng Assumptons and 3, Equaton 4) can be equvalently wrtten as: ẋ t) = A r x t) + B r c t) + B Λ u s t) + W T σ x t), x s t), c t)) + B δ j x j t)) +B Λ u s t) u t)) + B K x s t) x t)) 5) where x s t) R n s the sampled state vector, σ x t), x s t), c t)) β T x t)), x T s t), ct t) T R g +n +m. Now, let the adaptve feedback control law be gven by: C : u t) = Ŵ t) T σ x s t), c t)) 6)

7 Sensors 26, 6, of 3 where σ x s t), c t)) = β T x st)), x T s t), ct t) T R g +n +m, and Ŵ t) satsfes the weght update law: Ŵ t) = γ Proj m Ŵ t), σ x s t), c t)) e T s t)p B, Ŵ ) = Ŵ 7) wth e s t) x s t) x r t) R n beng the error of the trggered module state vector. Note that usng Equaton 6), Equaton 5) can be rewrtten as: ẋ t) =A r x t) + B r c t) B Λ W Tt)σ x s t), c t)) B Λ g ) + B δ j x j t)) + B Λ u s t) u t)) + B K x s t) x t)) 8) where g ) W T σ x s t), c t)) σ x t), x s t), c t)), and usng Equatons 8) and 6), we can wrte the module error dynamcs as: ė t) =A r e t) B Λ W Tt)σ x s t), c t)) B Λ g ) + B δ j x j t)) + B Λ u s t) u t)) + B K x s t) x t)) 9) The proposed event-trggered decentralzed adaptve control algorthm s based on the two-way data exchange structure depcted n Fgure, where the local controller generates u t) and the uncertan dynamcal module s drven by the sampled verson of ts local control sgnal u s t) dependng on an event-trggerng mechansm. Smlarly, the local controller utlzes x s t) that represents the sampled verson of the uncertan dynamcal module state x t) dependng on an event-trggerng mechansm. For ths purpose, let ɛ x R + be a gven, user-defned sensng threshold to allow for data transmsson from the uncertan dynamcal system to the controller. In addton, let ɛ u R + be a gven, user-defned actuaton threshold to allow for data transmsson from the local controller to the uncertan dynamcal module. Smlar n fashon to 33,35, we now defne three logc rules for schedulng the two-way data exchange: E : x s t) x t) ɛ x 2) E 2 : u s t) u t) ɛ u 2) E 3 : The controller receves x s t) 22) Specfcally, when the nequalty n Equaton 2) s volated at the s k moment of the k -th tme nstant, the uncertan module trggers the measured state sgnal nformaton, such that x s t) s sent to ts local controller. Lkewse, when Equaton 2) s volated or the local controller receves a new transmtted module state from the uncertan dynamcal system.e., when E 2 E 3 s true), then the local controller sends a new control nput u s t) to the uncertan dynamcal module at the r j moment of the j -th tme nstant. We now analyze the system stablty and performance of the proposed event-trggered decentralzed adaptve control algorthm ntroduced n ths secton usng the error dynamcs gven by Equaton 9), as well as the data exchange rules E, E 2, and E 3 respectvely gven by Equatons 2) 22). For organzatonal purposes, the rest of ths secton, s dvded nto four subsectons. Specfcally, we analyze the unform ultmate boundedness of the resultng closed-loop dynamcal system n Secton 2.2., compute the ultmate bound and hghlght the effect of user-defned thresholds and the adaptve controller desgn parameters on ths ultmate bound n Secton 2.2.2, show that the proposed archtecture does not yeld to a Zeno behavor n Secton and generalze the decentralzed event-trggered adaptve control algorthm usng a state emulator-based framework n Secton

8 Sensors 26, 6, of 3 Uncertan Large-Scale Module ZOH Event Trggerng Mechansm Adaptve Controller ZOH Fgure. Event-trggered adaptve control for large-scale modular systems Stablty Analyss and Unform Ultmate Boundedness We now present the frst result of ths paper, where the followng assumpton s needed. Assumpton 4. D λ mn R ) 2λ max P ) B F α j λ max P j ) B j F α j s postve by sutable selecton of the desgn parameters. Theorem. Consder the uncertan large-scale modular system S consstng of N nterconnected modules S descrbed by Equaton 4) subject to Assumptons 4. Consder, n addton, the reference model gven by Equaton 6), and the module feedback control law gven by Equatons 6) and 7). Moreover, let the data transmsson from the uncertan dynamcal module to the local controller occur when E s true and the data transmsson from the controller to the uncertan dynamcal system occur when E 2 E 3 s true. Then, the closed-loop soluton e t), W t)) s unformly ultmately bounded for all =, 2,..., N. Proof. Snce the data transmsson from the uncertan modules to ther local controllers and from the local controllers to the uncertan modules occur when E and E 2 E 3 are true, respectvely, note that x s t) x t) ɛ x and u s t) u t) ɛ u hold. Consder now the Lyapunov-lke functon gven by: ) V e, W ) = e T P e + γ tr W Λ 2 ) T 2 W Λ ) 23) Note that V, ) = and V e, W ) > for all e, W ) =, ). The tme-dervatve of Equaton 23) s gven by:

9 Sensors 26, 6, of 3 V e t), W t)) = 2e Tt)Pė t) + 2γ tr W Tt) W ) t)λ 2e T t)p A r e t) B Λ W Tt)σ x s t), c t)) B Λ g ) + B δ j x j t)) +B Λ u s t) u t)) + B K x s t) x t)) ) ) + 2tr W Tt)Λ σ x s t), c t)) es T t)p B e Tt)R e t) 2e Tt)P B Λ g ) + 2e Tt)P B δ j x j t)) + 2e Tt)P B Λ u s t) u t)) ) +2e Tt)P B K x s t) x t)) + 2tr W Tt)Λ σ x s t), c t)) x s t) x t)) T P B λ mn R ) e t) e t) λ max P ) B F Λ F g ) + 2e t)p B δ j x j t)) +2 e t) λ max P ) B F Λ F ɛ u + 2 e t) λ max P ) B F K F ɛ x + 2 W t) F Λ F 24) σ x s t), c t)) ɛ x λ max P ) B F It follows from Assumpton that an upper bound for g ) n Equaton 24) can be gven by: g ) = W T σ x s t), c t)) σ x t), x s t), c t)) Λ F ω L β }{{} x st) x t) K gɛ x 25) K g where K g R +. In addton, one can compute an upper bound for σ x s t), c t)) n Equaton 24) as: σ x s t), c t)) β x s t)) + x s t) + c t) L β x s t) + x s t) + c t) = L β + )ɛ x + L β + ) e t) + L β + )x r + c t) 26) where x r t) xr. Then, usng the bounds gven by Equatons 25) and 26) n Equaton 24), one can wrte: V e t), W t)) λ mn R ) e t) 2 + 2λ max P ) B F Λ F K g ɛ x + 2λ max P ) B F Λ F ɛ u + 2λ max P ) B F K F ɛ x + 2 W t) F Λ F L β + )λ max P ) B F ɛ x ) e t) + 2 W t) F Λ F L β + )ɛ x + L β + )xr + c t) ) λ max P ) B F ɛ x + 2e t)p B δ j x j t)) = c e t) 2 + c 2 e t) + c 3 + 2e t)p B δ j x j t)) 27) where c λ mn R ), c 2 2λ max P ) B F Λ F K g ɛ x + 2λ max P ) B F Λ F ɛ u + 2λ max P ) B F K F ɛ x + 2 w Λ F L β + )λ max P ) B F ɛ x and c 3 2 w Λ F Lβ + )ɛ x + L β + )xr + c t) ) λ max P ) B F ɛ x wth W t) F w due to utlzng the projecton operator n the weght update law gven by Equaton 9). Snce x j t) = e j t) + x rj t) wth x rj t) xrj, t follows from Assumpton 2 that:

10 Sensors 26, 6, of 3 δ j x j t)) α j e j t) + xrj 28) Furthermore, usng Equaton 28) n the last term of Equaton 27) results n: 2e t)p B δ j x j t)) 2λ max P ) e t) B F δ j x j t)) 2λ max P ) e t) B F α j e j t) + xrj λ max P ) B F α j 2 e t) e j t) + 2 e t) xrj λ max P ) B F α j 2 e t) 2 + e j t) 2 + xrj2 29) where Young s nequalty 46 s consdered n the scalar form of 2xy νx 2 + y 2 /ν, where x, y R and ν >, and appled to terms e t) e j t) and e t) xrj wth ν =. Hence, Equaton 27) becomes: V e t), W t)) c 2λ max P ) B F α j } {{ } d where ϕ c 3 + λ max P ) B F α j x rj2. Introducng: for the uncertan system S results n: e t) 2 + λ max P ) B }{{ F } f α j e j t) 2 + c 2 e t) + ϕ 3) V ) = N V e t), W t)) 3) = V ) = N = N = d e t) 2 + f α j e j t) 2 + c 2 e t) + ϕ ) e t) 2 + c 2 e t) + ϕ d f j α j } {{ } D 32) where D > s defned n Assumpton 4. Lettng e a t) e t),..., e N t) T, D dag D,..., D N ), C2 dag c 2,..., c 2N ) and ϕa N = ϕ, Equaton 32) can equvalently be wrtten as: V ) e T a t)d e a t) + C 2 e a t) + ϕ a λ mn D ) e a t) 2 + λ max C 2 ) e a t) + ϕ a 33) When e a t) > ψ, ths renders V ) <, where ψ are unformly ultmate bounded for all =, 2,..., N. λmaxc 2 ) 2 λ mn D ) + λ 2 max C 2 ) 4λ mn D ) +ϕ a λmn D ). Hence, e t) and W t)

11 Sensors 26, 6, 297 of Computaton of the Ultmate Bound for System Performance Assessment For revealng the effect of user-defned thresholds and the event-trggered feedback adaptve controller desgn parameters to the system performance, the next corollary presents a computaton of the ultmate bound for the system S. For ths purpose, we defne the followng, P mn dag λ mn P ),..., λ mn P N )), P max dag λ max P ),..., λ max P N ) ), ) γ a dag γ,..., γ N, Λ a dag Λ F,..., Λ N F ), W a t) T. W t) F,..., W N t) F Corollary. Consder the uncertan dynamcal system S consstng of N nterconnected modules S descrbed by Equaton 4) subject to Assumptons 4. Consder, n addton, the reference model gven by Equaton 6), and the module feedback control law gven by Equatons 6) and 7). Moreover, let the data transmsson from the uncertan modules to ther local controllers occur when E s true and the data transmsson from the controllers to the uncertan modules occur when E 2 E 3 s true. Then, the ultmate bound of the system error between the uncertan dynamcal system and the reference model s gven by: where: e a t) Φλ 2 mn P mn), t T 34) Φ λ max P max )ψ 2 + λ max γ a )λ max Λ a ) W a t) ) Proof. It follows from the proof of Theorem that Ve a t), W a t)) outsde the compact set gven by: S {e a t) : e a t) ψ} 36) That s, snce Ve a t), W a t)) cannot grow outsde S, the evoluton of Ve a t), W a t)) s upper bounded by: Ve a t), W a t)) max e a t) S Ve at), W a t)) It follows from e T a P mn e a Ve a, W a ) that e a t) 2 = λ max P max )ψ 2 + λ max γ a )λ max Λ a ) W a t) 2 = Φ 2 37) Φ 2, and Equaton 34) s mmedate. λ mn P mn ) Computaton of the Event-Trggered Inter-Sample Tme Lower Bound We now show that the proposed event-trggered decentralzed adaptve control archtecture does not yeld to a Zeno behavor, whch mples that t does not requre a contnuous two-way data exchange and reduces wreless network utlzaton. For the followng corollary presentng the result of ths subsecton, we consder r k q s k, s k +) to be the q -th tme nstant when E 2 s volated over s k, s k +), and snce { } s k k = s a subsequence of { } r j j =, t follows that { } r j j = = { } { } s k r k,mk k = q, k =,q = where m k N s the number of volaton tmes of E 2 over s k, s k +). Corollary 2. Consder the uncertan dynamcal system S consstng of N nterconnected modules S descrbed by Equaton 4) subject to Assumptons 4. Consder, n addton, the reference model gven by Equaton 6), and the module feedback control law gven by Equatons 6) and 7). Moreover, let the data transmsson from the uncertan dynamcal module to the local controller occur when E s true and the data transmsson from the

12 Sensors 26, 6, 297 of 3 controller to the uncertan dynamcal system occur when E 2 E 3 s true. Then, there exst postve scalars α x ɛ x Φ and α u ɛ u Φ such that: 2 s k + s k > α x, k N 38) r k q + rk q > α u, q {,..., m k }, k N 39) Proof. The tme dervatve of x s t) x t) over t s k, s k +), k N, s gven by: d dt x st) x t) ẋ s t) ẋ t) = ẋ t) A r F e t) + xr + B r F c t) + B F Λ F w L β ɛx + e t) +xr) + K F ɛ x + e t) + xr ) + K 2 F c t) + B F Λ F K g ɛ x + B F α j e j t) + xrj ) + B F Λ F ɛ u + B F K F ɛ x 4) Snce the closed-loop dynamcal system s unformly ultmately bounded by Theorem, there exsts an upper bound to Equaton 4). Lettng Φ denote ths upper bound and wth the ntal condton satsfyng lm t s + k x s t) x t) =, t follows from Equaton 4) that: x s t) x t) Φ t s k ), t s k, s k +) 4) Therefore, when E s true, then lm t s x s t) x t) = ɛ x, and t then follows from Equaton 4) k + that s k + s k α x. ) Smlarly, the tme dervatve of u s t) u t) over t r k q, r k q +, q N, s gven by: d dt u st) u t) u s t) u t) = u t) = Ŵ Tt)σ x s t), c t)) + Ŵ Tt) σ x s t), c t)) γ B F λ max P ) e s t) σ x s t), c t)) 2 + Λ F K 2 F ċ t) γ B F λ max P ) e t) + ɛ x ) L β ɛ x + e t) + xr ) + K F ɛ x + e t) + x r ) + K 2 F c t) 2 + Λ F K 2 F ċ t) 42) Once agan, snce the closed-loop dynamcal system s unformly ultmately bounded by Theorem, there exsts an upper bound to Equaton 42). Lettng Φ 2 denote ths upper bound, and wth the ntal condton satsfyng lm k t r + u s t) u t) =, t follows from Equaton 42) that: q u s t) u t) Φ 2 t r k q ), t ) r k q, r k q + Therefore, when Ē 2 E 3 s true, then lm k t r u s t) u t) = ɛ u, and t then follows from q + Equaton 43) that r k q + rk q α u. 43) Corollary 2 shows that the nter-sample tmes for the module state vector and decentralzed feedback control vector are bounded away from zero, and hence, the proposed event-trggered adaptve control approach does not yeld to a Zeno behavor. As dscussed earler, ths mples that the proposed event-trggered decentralzed adaptve control methodology does not requre a contnuous two-way data exchange, and t reduces wreless network utlzaton.

13 Sensors 26, 6, of Generalzatons to the Event-Trggered Decentralzed Adaptve Control wth State Emulator We now generalze our framework to a state emulator-based desgn, snce ths framework has the capablty to suppress possble hgh-frequency oscllaton n the control sgnal of the uncertan module S,3, Consder the modfed) reference system, so-called the state emulator of S, gven by: ˆx t) = A r ˆx t) + B r c t) + L x s t) ˆx t)), ˆx ) = ˆx 44) where L R n n + D n n s the state emulator gan. Lettng ê t) ˆx t) x r t) R n, the reference model error dynamcs capturng the dfference between the deal reference model n Equaton 6) and the state emulator-based modfed) reference model n Equaton 44) s gven by: ê t) = A r ê t) + L x s t) ˆx t)) 45) In addton, lettng x t) x t) ˆx t) R n to denote the system state error vector, the state emulator-based) system error dynamcs follows from Equatons 8) and 44) as: x t) = A L x t) B Λ W T t)σ x s t), c t)) B Λ g ) + B δ j x j t)) + B Λ u s t) u t)) +B K L )x s t) x t)), x ) = x 46) where A L A r L R n n s Hurwtz by a sutable selecton of the state emulator gan L e.g., A L s Hurwtz wth L = κ I, κ R +, snce A r s Hurwtz). To mantan system stablty, we utlze the adaptve controller gven by Equaton 6) wth the update law descrbed by: Ŵ t) γ Proj m Ŵ t), σ x s t), c t)) x s t) ˆx t)) T P B, Ŵ ) = Ŵ 47) where P R n n S n n + s the unque soluton of the algebrac Rccat equaton: = A T L P + P A L P B R B T P + Q 48) wth R R m m + S n n and Q R n n + S n n. Note from,42 that the state emulator-based adaptve control framework acheves strngent transent and steady-state system performance specfcatons by judcously choosng the learnng rate γ and the state emulator gan L wthout causng hgh-frequency oscllatons n the controller response, unlke standard model reference adaptve controllers overvewed earler n ths secton. We also note that f one selects L =, then the results of ths paper hold for standard model reference adaptve controllers, and hence, there s no loss n generalty n usng a state emulator-based adaptve control framework for the man results of ths paper. Consder a parameter-dependent Rccat equaton 23,47 gven by: = A T r P + P A r + Q 49) Q = µ P L L T P + Q o, 5) where P R n n + s a unque soluton wth Q o R n n + and µ >. Remark 23. Let < µ < µ defne the largest set wthn whch there s a postve-defnte soluton for P. Snce P > for µ = and P > depends contnuously on µ, the exstence of P µ ) > for < µ < µ s assured. The next lemma shows that for µ < µ, Equatons 49) and 5) can relably be solved for P > usng the Potter approach gven n 48. Ths also mples that µ can be determned by searchng for the boundary value, µ. We employ notaton rc ) and dom ) as defned n 48.

14 Sensors 26, 6, of 3 Lemma 23,48. Let P > satsfy the parameter dependent Rccat equaton gven by Equatons 49) and 5), and let the modfed Hamltonan be gven by: H = A r Q o µ L L T Then, for all < µ < µ, H domrc) and P = rch ). Assumpton 5. D λ mn Q ) λ mn R )λ 2 maxp ) B 2 F l µ 3λ max P ) B F α j λ max P j ) B j F α j and D 2 l λ mn Q o ) λ max P j ) B j F α j, l >, are postve by sutable selecton of the desgn parameters. Corollary 3. Consder the uncertan dynamcal system S consstng of N nterconnected modules S descrbed by Equaton 4) subject to Assumptons 3 and 5. Consder n addton, the deal reference model gven by Equaton 6), the state emulator gven by Equaton 44) and the module feedback control law gven by Equatons 6) and 47). Moreover, let the data transmsson from the uncertan dynamcal module to the local controller occur when E s true and the data transmsson from the controller to the uncertan dynamcal system occur when E 2 E 3 s true. Then, the closed-loop soluton x t), W t), ê t)) s unformly ultmately bounded for all =, 2,..., N. Proof. Consder the Lyapunov-lke functon gven by: A T r V x, W, ê ) = x T P x + γ tr W Λ 2 ) T W Λ 2 ) + l ê T 5) P ê 52) where l > and P > satsfes the parameter dependent Rccat equaton n Equatons 49) and 5). Note that V,, ) = and V x, W, ê ) > for all x, W, ê ) =,, ). The tme-dervatve of Equaton 52) s gven by: V x t), W t), ê t)) = 2 x Tt)P x t) + 2γ tr W t)λ 2 ) T W 2 t)λ ) + 2l ê Tt) P ê t) 2 x Tt)P A L x t) B Λ W Tt)σ x s t), c t)) B Λ g ) + B δ j x j t)) + B Λ us t) u t) ) +B K L )x s t) x t)) + 2tr W Tt)σ x s t), c t)) x s t) ˆx t)) T P B Λ +2l ê Tt) P Ar ê t) + L x s t) ˆx t)) x Tt)Q x t) + x Tt)P B R B T P x t) 2 x Tt)P B Λ g ) + 2 x Tt)P B δ j x j t)) +2 x Tt)P B Λ us t) u t) ) + 2 x Tt)P B K L )x s t) x t)) + 2tr W t) T 53) σ x s t), c t)) x s t) x t)) T P B Λ l ê Tt) Q o ê t) l ê Tt)µ P L L T P ê t) +2l ê T t) P L x s t) x t)) + 2l ê T t) P L x t) Young s nequalty 46 appled to the last term n Equaton 53) produces: 2l ê T t) P L x t) µ l ê T t) P L L T P ê t) + l µ x T t) x t) 54)

15 Sensors 26, 6, of 3 Usng Equaton 54) n Equaton 53) yelds: V x t), W t), ê t)) x Tt)Q x t) + x Tt)P B R B T P x t) 2 x Tt)P B Λ g ) + 2 x Tt)P B δ j x j t)) +2 x Tt)P B Λ us t) u t) ) + 2 x Tt)P B K L )x s t) x t)) + 2tr W Tt)σ x s t), c t)) x s t) x t)) T P B Λ l ê T t) Q o ê t) + 2l ê T t) P L x s t) x t)) + l µ x T t) x t) λ mn R ) x t) 2 + λ mn R )λ 2 maxp ) B 2 F x t) 2 + 2λ max P ) B F Λ F g ) x t) + 2 x t)p B δ j x j t)) + 2 x t) λ max P ) B F Λ F ɛ u + 2 x t) λ max P ) B K F + L F ) ɛx + 2 W t) F σ x s t), c t)) ɛ x λ max P ) B F Λ F l λ mn Q o ) ê t) 2 55) +2l ê t) λ max P ) L F ɛ x + l µ x t) 2 Usng Equatons 25) and 26), Equaton 55) can be wrtten: V x t), W t), ê t)) λ mn Q ) λ mn R )λ 2 maxp ) B 2 F l µ ) x t) 2 l λ mn Q o ) ê t) 2 + 2λ max P ) B F Λ F K g ɛ x + 2λ max P ) B F Λ F ɛ u + 2λ max P ) B F K F ɛ x + 2 W t) F Λ F L β + )λ max P ) B F ɛ x ) x t) + 2 W t) F Λ F Lβ + )ɛ x + L β + )x r + c t) ) λ max P ) B F ɛ x + 2 x t)p B δ j x j t)) + 2l λ max P ) L F ɛ x ê t) = c x t) 2 c 2 ê t) 2 + c 3 x t) + c 4 ê t) + c x t)p B δ j x j t)) 56) where c λ mn Q ) λ mn R )λ 2 maxp ) B 2 F l µ, c 2 l λ mn Q o ), c 3 2λ max P ) B F Λ F K g ɛ x + 2λ max P ) B F Λ F ɛ u + 2λ max P ) B F K F ɛ x + 2 w Λ L β + )λ max P ) B F ɛ x, c 4 2l λ max P ) L F ɛ x and c 5 2 w Λ F Lβ + )ɛ x + L β + )x r + c t) ) λ max P ) B F ɛ x. Snce x j t) = x j t) + ê j t) + x rj t), t follows from Assumpton 2 that: δ j x j t)) α j x j t) + ê j t) + xrj Furthermore, usng Equaton 57) n the last term of Equaton 56) results n: 2 x t)p B δ j x j t)) 2λ max P ) x t) B F δ j x j t)) 2λ max P ) x t) B F α j x j t) + ê j t) + xrj λ max P ) B F α j 2 x t) x j t) + 2 x t) ê j t) + 2 x t) xrj λ max P ) B F α j 3 x t) 2 + x j t) 2 + ê j t) 2 + xrj2 58) where Young s nequalty 46 s consdered n the scalar form of 2xy νx 2 + y 2 /ν, wth x, y R and ν >, and appled to terms x t) x j t), x t) ê j t) and x t) xrj wth ν =. Hence, Equaton 56) becomes: 57)

16 Sensors 26, 6, of 3 V x t), W t), ê t)) c 3λ max P ) B F α j } {{ } d + λ max P ) B }{{ F } f x t) 2 c 2 ê t) 2 + c 3 x t) + c 4 ê t) α j x j t) 2 + λ max P ) B }{{ F } f where ϕ c 5 + λ max P ) B F α j x rj2. Introducng: V ) = for the uncertan system S results n: α j ê j t) 2 + ϕ 59) N V x t), W t)ê t)) 6) = V ) N d x t) 2 c 2 ê t) 2 + c 3 x t) = + c 4 ê t) + f = N = d f j α j }{{} D + c 3 x t) + c 4 ê t) + ϕ α j x j t) 2 + f α j ê j t) 2 + ϕ ) x t) 2 c 2 f j α j } {{ } D 2 ) ê t) 2 6) where D > and D 2 > are defned n Assumpton 5. Lettng x a t) x t),..., x N t) T, ê a t) ê t),..., ê N t) T, D dag D,..., D N ), D2 dag D 2,..., D 2N ), C3 dag c 3,..., c 3N ), C4 dag c 4,..., c 4N ) and ϕa N = ϕ, then Equaton 6) can equvalently be wrtten as: V ) x T a t)d x a t) ê T a t)d 2 ê a t) + C 3 x a t) + C 4 ê a t) + ϕ a λ mn D ) x a t) 2 λ mn D 2 ) ê a t) 2 + λ max C 3 ) x a t) + λ max C 4 ) ê a t) + ϕ a 62) Ether x a t) > ψ or ê a t) > ψ 2 renders V ) <, where ψ and ψ 2 λmaxc 4 ) 2 λ mn D 2 ) + λ 2 max C 3 ) 4λ mn D ) + λ2 max C 4 ) 4λ mn D 2 ) +ϕ a λmn D 2 ) bounded for all =, 2,..., N. λmaxc 3 ) 2 λ mn D ) + λ 2 max C 3 ) 4λ mn D ) + λ2 max C 4 ) 4λ mn D 2 ) +ϕ a λmn D ), and hence, x t), ê t) and W t) are unformly ultmate Corollary 4. Under the condtons of Corollary 3, we can show that e t) s bounded for all =, 2,..., N. Proof. It readly follows from: e t) = x t) ˆxt) + ˆxt) x r t) x t) ˆxt) + ˆxt) x r t) and Corollary 3 that e t) s bounded for all =, 2,..., N. x t) + ê t) 63)

17 Sensors 26, 6, of 3 Remark 2. In order to obtan the closed-loop system error ultmate bound value for Equaton 63) and the no Zeno behavor characterzaton, we can follow the same steps hghlghted n Corollares and 2, respectvely. 3. Event-Trggered Dstrbuted Adaptve Control We now ntroduce an event-trggered dstrbuted adaptve control archtecture n ths secton, where t s assumed that physcally-nterconnected modules can locally communcate wth each other for exchangng ther state nformaton. For organzatonal purposes, ths secton s broken up nto two subsectons. Specfcally, we frst brefly overvew a standard dstrbuted adaptve control archtecture wthout event-trggerng and then present the proposed event-trggered decentralzed adaptve control approach, whch ncludes rgorous stablty and performance analyses wth no Zeno behavor and generalzatons to the state emulator case for suppressng the effect of possble hgh-frequency oscllatons n the controller response. As shown, the beneft of usng the proposed dstrbuted adaptve control archtecture versus the decentralzed archtecture of the prevous secton s that there s no need for any structural assumptons; that s, Assumptons 4 and 5, n the dstrbuted case to guarantee overall system stablty for applcatons where modules are allowed to locally communcate wth each other). 3.. Overvew of a Standard Dstrbuted Adaptve Control Archtecture wthout Event-Trggerng The standard dstrbuted adaptve control archtecture overvewed n ths secton bulds on the problem formulaton stated n Secton 2. wth an mportant dfference that the physcally-nterconnected modules can locally communcate wth each other for exchangng ther state nformaton, as dscussed above. For ths purpose, we frst replace Assumpton 2 of Secton 2. wth the followng assumpton. Assumpton 6. The functon δ j x j t)) n Equaton 2) satsfes: δ j x j t)) = Q T j φ jx j t)) 64) where Q j R g m s an unknown weght matrx and φ j : R n j R g s a known Lpschtz contnuous bass functon vector satsfyng: wth L φj R +. φ j x j ) φ j x 2j ) L φj x j x 2j 65) Remark 3. We can equvalently represent Equaton 64) as: Q T j φ jx j t)) Gj T F jx j t)) 66) where G j R g d ) m s the matrx combnaton for the deal weght matrces of the connected graph, F j x j t)) : R n j R g d ) s the vector combnaton for bass functon vectors of the connected graph and d s the degree of the -th agent. The rght hand sde of Equaton 66) can be gven as: G T j F jx j t)) = G T daga )F 67) where G R g N) m s the matrx combnaton for all modules deal weght matrces of the system toward S, F x j t)) : R n j R g N) s the vector combnaton for all bass functon vectors of the system toward S and A s the -th row of the adjacency matrx A.

18 Sensors 26, 6, of 3 Next, usng Assumptons, 3 and 6, Equaton 2) can be equvalently wrtten as: where W ẋ t) = A r x t) + B r c t) + B Λ u t) + W T σ x t), c t), x j t) ) 68) Λ W T o, Λ K T, Λ weght matrx and σ x t), c t), x j t) ) T K2 T, Λ Gj T R g +n +m +g d )) m s an unknown T β T x t)), x Tt), ct t), FT j jt)) x R g +n +m +g d ). Motvated from the structure of the uncertan terms appearng n Equaton 68), let the dstrbuted adaptve feedback controller of S, V G, be gven by: C : u t) = Ŵ t) T σ x t), c t), x j t) ) 69) where Ŵ t) s an estmate of W satsfyng the update law: Ŵ t) = γ Proj m Ŵ t), σ x t), c t), x j t) ) e T t)p B, Ŵ ) = Ŵ 7) where P R n n + S n n s a soluton of the Lyapunov Equaton ). Now, from Equatons 6) and 68), the module-level closed-loop error dynamcs can be gven by: ė t) = A r e t) B Λ W Tt)σ x t), c t), x j t) ), e t) = e 7) 3.2. Proposed Event-Trggered Dstrbuted Adaptve Control Archtecture We now present the proposed event-trggered dstrbuted adaptve control archtecture for modular systems, where each uncertan module can exchange ts state nformaton wth ts nterconnected neghborng modules. Consder the uncertan dynamcal module gven by: S : ẋ t) = A x t) + B Λ u s t) + x t)) + δ j x sj t)), x ) = x 72) where δ j x sj t)) Q T j φ jx sj t)) and x sj t) R n j. Usng Assumptons, 3 and 6, Equaton 72) can be equvalently wrtten as: ẋ t) =A r x t) + B r c t) + B Λ u s t) + W T σ x t), x s t), c t), x sj t) ) + B Λ u s t) u t)) + B K x s t) x t)) 73) T where σ x t), x s t), c t), x sj t) ) β T x t)), xs T t), ct t), FT j sjt)) x R g +n +m +g d ), and the dstrbuted adaptve feedback control s gven by: where σ xs t), c t), x sj t) ) satsfes the weght update law: C : u t) = Ŵ t) T σ xs t), c t), x sj t) ) 74) T β T x st)), xs T t), ct t), FT j sjt)) x R g +n +m +g d, and Ŵ t) Ŵ t) = γ Proj m Ŵ t), σ xs t), c t), x sj t) ) e T s t)p B, Ŵ ) = Ŵ 75) Now, usng Equaton 74) n Equaton 73) yelds: ẋ t) =A r x t) + B r c t) B Λ W Tt)σ xs t), c t), x sj t) ) B Λ g ) + B Λ u s t) u t)) + B K x s t) x t)) 76)

19 Sensors 26, 6, of 3 where g ) W T σ xs t), c t), x sj t) ) σ x t), x s t), c t), x sj t) ), and usng Equatons 76) and 6), we can wrte the module error dynamcs as: ė t) =A r e t) B Λ W Tt)σ xs t), c t), x sj t) ) B Λ g ) + B Λ u s t) u t)) + B K x s t) x t)) 77) For organzatonal purposes, we now dvde ths secton nto four sectons. Specfcally, we analyze the unform ultmate boundedness of the resultng closed-loop dynamcal system n Secton 3.2., compute the ultmate bound n Secton 3.2.2, show that the proposed archtecture does not yeld to a Zeno behavor n Secton and generalze the dstrbuted event-trggered adaptve control algorthm usng the state emulator-based framework n Secton Stablty Analyss and Unform Ultmate Boundedness Theorem 2. Consder the uncertan dynamcal system S consstng of N nterconnected modules S descrbed by Equaton 72) subject to Assumptons, 3 and 6. Consder, n addton, the reference model gven by Equaton 6) and the module feedback control law gven by Equatons 74) and 75). Moreover, let the data transmsson from the uncertan dynamcal module to the local controller occur when E s true and the data transmsson from the controller to the uncertan dynamcal system occur when E 2 E 3 s true. Then, the closed-loop soluton e t), W t)) s unformly ultmately bounded for all =, 2,..., N. Proof. Snce the data transmsson from the uncertan dynamcal module to the local controller and from the local controller to the uncertan dynamcal module occur when E and E 2 E 3 are true, respectvely, note that x s t) x t) ɛ y and u s t) u t) ɛ u hold. Consder the Lyapunov-lke functon gven by: ) V e, W ) = e T P e + γ tr W Λ 2 ) T 2 W Λ ) 78) Note that V, ) = and V e, W ) > for all e, W ) =, ). The tme dervatve of Equaton 78) s gven by: V e t), W t)) = 2e Tt)Pė t) + γ 2tr W Tt) W ) t)λ 2e Tt)P A r e t) B Λ W Tt)σ xs t), c t), x sj t) ) B Λ g ) + B Λ u s t) u t)) +B K x s t) x t)) ) + 2tr W Tt)Λ σ xs t), c t), x sj t) ) ) es T t)p B e Tt)R e t) 2e Tt)P B Λ g ) + 2e Tt)P B Λ u s t) u t)) + 2e Tt)P B K x s t) x t)) +2tr W Tt)Λ σ xs t), c t), x sj t) ) ) x s t) x t)) T P B λ mn R ) e t) e t) λ max P ) B F Λ F g ) + 2 e t) λ max P ) B F Λ F ɛ u + 2 e t) λ max P ) B F K F ɛ x + 2 W t) F Λ F σ xs t), c t), x sj t) ) 79) ɛ x λ max P ) B F where the same upper bound g ) has the same result of Equaton 25). In addton, one can compute an upper bound for σ xs t), c t), x sj t) ) n Equaton 79) as:

20 Sensors 26, 6, of 3 σ xs t), c t), x sj t) ) β x s t)) + x s t) + c t) + F j x sj t)) L β x s t) + x s t) + c t) + φ j x j t)) = L β + )ɛ x + L β + ) e t) + L β + )xr + c t) ) + L φj ɛ xj + e j t) + xrj 8) where x r t) xr and x rjt) xrj. Then, usng the bounds gven by Equatons 25) and 8) n Equaton 79) yelds: V e t), W t)) λ mn R ) e t) e t) λ max P ) B F Λ F K g ɛ x + 2 e t) λ max P ) B F Λ F ɛ u +2 e t) λ max P ) B F K F ɛ x + 2 W t) F Λ F L β + )ɛ x + L β + ) e t) +L β + )xr + c t) + L φj ɛxj + e j t) + xrj) ) ɛ x λ max P ) B F λ mn R ) e t) 2 + 2λ max P ) B F Λ F K g ɛ x + 2λ max P ) B F Λ F ɛ u +2λ max P ) B F K F ɛ x + 2 w Λ F L β + )ɛ x λ max P ) B F ) e t) +2 w Λ F L β + )ɛ x + L β + )xr + c t) + L φj ɛxj + xrj) ) ɛ x λ max P ) B F +2 w Λ F ɛ x λ max P ) B F L φj e j t) d e t) 2 + d 2 e t) + d 3 + f L φj e j t) 8) where d λ mn R ), d 2 2λ max P ) B F Λ F K g ɛ x + 2λ max P ) B F Λ F ɛ u + 2λ max P ) B F K F ɛ x +2 w Λ F L β + )ɛ x λ max P ) B F, d 3 2 w Λ F L β + )ɛ x + L β + )x r + c t) + L φj ɛxj + x rj) ) ɛ x λ max P ) B F and f 2 w Λ F ɛ x λ max P ) B F. Introducng: for the uncertan system S results n: V ) = N V e t), W t)) 82) = V ) = N = N = d e t) 2 + d 2 e t) + f L φj e j t) + d 3 d e t) 2 + d 2 + f j L φj ) e t) + d 3 } {{ } D 2 83) where D >. Lettng e a t) e t),..., e N t) T, D dag d,..., d N ), D 2 dag D 2,..., D 2N ), and D3 N = d 3, then Equaton 32) can equvalently be wrtten as:

21 Sensors 26, 6, of 3 V ) e T a t)d e a t) + D 2 e a t) + D 3 When e a t) > ψ, ths renders V ) <, where ψ λ mn D ) e a t) 2 + λ max D 2 ) e a t) + D 3 84) W t) are unformly ultmate bounded for all =, 2,..., N. λmaxd 2 ) 2 λ mn D ) + λ 2 max D 2 ) 4λ mn D ) +D 3 λmn D ) Computaton of the Ultmate Bound for System Performance Assessment, and hence, e t) and For revealng the effect of user-defned thresholds and the event-trggered output feedback adaptve controller desgn parameters to the system performance, the next corollary presents a computaton of the ultmate bound. Corollary 5. Consder the uncertan dynamcal system S consstng of N nterconnected modules S descrbed by Equaton 72) subject to Assumptons, 3 and 6. Consder, n addton, the reference model gven by Equaton 6) and the module feedback control law gven by Equatons 74) and 75). Moreover, let the data transmsson from the uncertan dynamcal module to the local controller occur when E s true and the data transmsson from the controller to the uncertan dynamcal system occur when E 2 E 3 s true. Then, the ultmate bound of the system error between the uncertan dynamcal system and the reference model s gven by: where e a t) Φλ 2 mn P mn), t T 85) Φ λ max P max )ψ 2 + λ max γ a )λ max Λ a ) W a t) ) Proof. The proof s smlar to the proof of Corollary, and hence, omtted Computaton of the Event-Trggered Inter-Sample Tme Lower Bound In ths subsecton, we show that the proposed event-trggered dstrbuted adaptve control archtecture does not yeld to a Zeno behavor, whch mples that t does not requre a contnuous two-way data exchange and reduces wreless network utlzaton. For ths purpose, we use the same mathematcal notatons ntroduced n Secton and make the followng assumpton. Assumpton 7. Each module S holds the receved trggered state nformaton δ j x sj t)) from ts nterconnected neghborng modules S j and sends ths nformaton to ts local controller C when the condton E n Equaton 2) s volated. Corollary 6. Consder the uncertan dynamcal system S consstng of N nterconnected modules S descrbed by Equaton 72) subject to Assumptons, 3, 6 and 7. Consder, n addton, the reference model gven by Equaton 6) and the module feedback control law gven by Equatons 74) and 75). Moreover, let the data transmsson from the uncertan dynamcal module to the local controller occur when E s true and the data transmsson from the controller to the uncertan dynamcal system occur when E 2 E 3 s true. Then, there exst postve scalars α x ɛ x Φ and α u ɛ u Φ, such that: 2 s k + s k > α x, k N 87) r k q + rk q > α u, q {,..., m k }, k N 88)

22 Sensors 26, 6, of 3 Proof. The proof s smlar to the proof of Corollary 2, and hence, omtted. Corollary 6 also shows that the nter-sample tmes for the module state vector and dstrbuted feedback control vector are bounded away from zero, and hence, the proposed event-trggered dstrbuted adaptve control approach does not yeld to a Zeno behavor Generalzatons to the Event-Trggered Dstrbuted Adaptve Control wth State Emulator Smlar to Secton 2.2.4, consder the modfed) reference model, so-called the state emulator, gven by Equaton 44) and the reference model error dynamcs capturng the dfference between the deal reference model Equaton 6), and the state emulator-based modfed) reference model Equaton 44) s gven by Equaton 45). In addton, the state emulator-based) system error dynamcs follow from Equatons 76) and 44) as: x t) = A r x t) B Λ W Tt)σ xs t), c t), x sj t) ) B Λ g ) +B Λ u s t) u t)) + B K L )x s t) x t)) L x t), x ) = x 89) where the adaptve controller Equaton 74) s used and the weght update law s gven by: Ŵ t) = γ Proj m Ŵ t), σ xs t), c t), x sj t) ) x s t) ˆx t)) T P B, Ŵ ) = Ŵ 9) wth P R n n + S n n beng a soluton to the Lyapunov Equaton ). Corollary 7. Consder the uncertan dynamcal system S consstng of N nterconnected modules S descrbed by Equaton 72) subject to Assumptons, 3 and 6. Consder, n addton, the deal reference model gven by Equaton 6), the state emulator gven by Equaton 44) and the module feedback control law gven by Equatons 74) and 9). Moreover, let the data transmsson from the uncertan dynamcal module to the local controller occur when E s true and the data transmsson from the controller to the uncertan dynamcal system occur when E 2 E 3 s true. Then, the closed-loop soluton x t), W t), ê t)) s unformly ultmately bounded for all =, 2,..., N. Proof. Consder the Lyapunov-lke functon gven by: V x, W, ê ) = x T P x + γ tr W Λ 2 ) T W Λ 2 ) + 2l L F λ maxp )λ max R )ê T P ê 9) Note that V,, ) = and V x, W, ê ) > for all x, W, ê ) =,, ). The tme-dervatve of Equaton 9) s gven by: V x t), W t), ê t)) = 2 x Tt)P x t) + 2γ tr W t)λ 2 ) T W 2 t)λ ) + 4l L F λ maxp )λ mn R )ê T P ê t) 2 x Tt)P A r x t) B Λ W Tt)σ xs t), c t), x sj t) ) B Λ g ) + B Λ us t) u t) ) +B K L )x s t) x t)) L x t) + 2tr W Tt)σ xs t), c t), x sj t) ) x s t) ˆx t)) T P B Λ +4l L F λ maxp )λ mn R )ê Tt)P Ar ê t) + L x t)) + L x s t) x t)) x Tt)R x t) 2 x Tt)P B Λ g ) + 2 x Tt)P B Λ us t) u t) ) + 2 x Tt)P B K L ) x s t) x t)) 2 x Tt)P L x t) + 2tr W t) T σ xs t), c t), x sj t) ) x s t) x t)) T P B Λ l L F λ maxp )λ mn R )ê Tt)R ê t) + 4l L F λ maxp )λ mn R )ê Tt)P L x s t) x t)) +4l L F λ maxp )λ mn R )ê Tt)P L x t)

23 Sensors 26, 6, of 3 λ mn R ) x t) 2 + 2λ max P ) B F Λ F g ) x t) + 2 x t) λ max P ) B F Λ F ɛ u +2 x t) λ max P ) ) B K F + L F ɛx 2λ max P ) L x t) W t) F σ xs t), c t), x sj t) ) λ max P ) B F Λ F ɛ x 2l L F λ maxp )λ 2 mn R ) ê t) 2 +4l λ mn R )ɛ x ê t) + 4l λ mn R ) ê t) x t) 92) Now, usng Young s nequalty 46 for the last term n Equaton 92), wth µ R +, yelds: V x t), W t), ê t)) λ mn R ) x t) 2 + 2λ max P ) B F Λ F g ) x t) + 2 x t) λ max P ) B F Λ F ɛ u + 2 x t) λ max P ) ) B K F + L F ɛx 2λ max P ) L x t) W t) F σ xs t), c t), x sj t) ) λ max P ) B F Λ F ɛ x 2l L F λ maxp )λ 2 mn R ) ê t) 2 + 4l λ mn R )ɛ x ê t) + 2l µ λ mn R ) ê t) l µ λ mn R ) x t) 2 93) Usng Equatons 25) and 79), Equaton 93) can be wrtten by: V x t), W t), ê t)) λ mn R ) 2λ max P ) L F 2 l λ mn R µ ) x t) 2 2 l L F λ maxp )λ 2 mn R ) l µ λ mn R ) ê t) 2 + 2λ max P ) B F Λ F K g ɛ x + 2λ max P ) B F Λ F ɛ u +2λ max P ) ) B K F + L F ɛx x t) + 4l λ mn R )ɛ x ê t) +2 w L β + )ɛ x + L β + ) x t) + ê t) + L β + )xr + c t) + L φj ɛxj + x j t) + ê j t) + xrj) λ max P ) B F Λ F ɛ x λ mn R ) 2λ max P ) L F 2 l λ mn R µ ) x t) 2 2 l L F λ maxp )λ 2 mn R ) l µ λ mn R ) ê t) 2 + 2λ max P ) B F Λ F K g ɛ x + 2λ max P ) B F Λ F ɛ u +2λ max P ) ) B K F + L F ɛx + 2 w λ maxp ) B F Λ F ɛ x x t) + 4l λ mn R )ɛ x + 2 w λ maxp ) B F Λ F ɛ x ê t) +2 w λ maxp ) B F Λ F ɛ x L β + )ɛ x + xr ) + c t) + L φj ɛxj + x ) ) rj +2 w λ maxp ) B F Λ F ɛ x L φj xj t) + ê j t) ) 94) then settng µ = l λ mn R )λmaxp ) L F n Equaton 94) yelds: V x t), W t), ê t)) λ mn R ) x t) 2 2l L F λ maxp )λ mn R ) λ mn R ) l ê t) 2 + 2λ max P ) B F Λ F K g ɛ x + 2λ max P ) B F Λ F ɛ u + 2λ max P ) ) B K F + L F ɛx 4l λ mn R )ɛ x + 2 w λ maxp ) B F Λ F ɛ x ê t) + 2 w λ maxp ) B F Λ F ɛ x x t) w λ maxp ) B F Λ F ɛ x L β + )ɛ x + xr ) + c t) + L φj ɛxj + x ) ) rj + 2 w λ maxp ) B F Λ F ɛ x L φj xj t) + ê j t) ) 95)

24 Sensors 26, 6, of 3 It then follows that Equaton 95) can be gven by: V x t), W t), ê t)) d x t) 2 d 2 ê t) 2 + d 3 x t) + d 4 ê t) + d 5 + f L φj x j t) + f L φj ê j t) 96) where d λ mn R ), d 2 2l L F λ maxp )λ mn R ) λ mn R ) l, d 3 2λ max P ) B F Λ F K g ɛ x +2λ max P ) B F Λ F ɛ u + 2λ max P ) ) B K F + L F ɛx + 2 w λ maxp ) B F Λ F ɛ x, d 4 4l λ mn R ) ɛ x + 2 w λ maxp ) B F Λ F ɛ x, d 5 2 w λ maxp ) B F Λ F ɛ x L β + )ɛ x + x r ) + c t) + L φj ɛxj + x rj) ) and f 2 w λ maxp ) B F Λ F ɛ x. To ensure that d 2 s postve defnte, we consder l = θ λ mn R ) and θ, ). Introducng: V ) = for the uncertan system S results n: V ) N = N V x t), W t)ê t)), 97) = d x t) 2 d 2 ê t) 2 + d 3 x t) + d 4 ê t) + d 5 + f L φj x j t) + f L φj ê j t) = N = d x t) 2 d 2 ê t) 2 + d 3 + f j L φj } {{ } D 3 ) x t) + d 4 + f j L φj } {{ } D 4 ) ê t) + d 5 98) Lettng x a t) x t),..., x N t) T, êa t) ê t),..., ê N t) T, D dag d,..., d N ), D2 dag d 2,..., d 2N ), D3 dag D 3,..., D 3N ), D4 dag D 4,..., D 4N ), and D5 N = d 5, then Equaton 98) can equvalently be wrtten as: V ) x T a t)d x a t) ê T a t)d 2 ê a t) + D 3 x a t) + D 4 e a t) + D 5 λ mn D ) x a t) 2 λ mn D 2 ) ê a t) 2 + λ max D 3 ) x a t) + λ max D 4 ) ê a t) + D 5 99) Ether x a t) > ψ or ê a t) > ψ 2, renders V ) <, where ψ and ψ 2 λmaxd 4 ) 2 λ mn D 2 ) + λ 2 max D 3 ) 4λ mn D ) + λ2 max D 4 ) 4λ mn D 2 ) +D 5 λmn D 2 ) bounded for all =, 2,..., N. λmaxd 3 ) 2 λ mn D ) + λ 2 max D 3 ) 4λ mn D ) + λ2 max D 4 ) 4λ mn D 2 ) +D 5 λmn D ), and hence, x t), ê t), and W t) are unformly ultmate Remark 4. To show that e t) s bounded for all =, 2,..., N under the condton of Corollary 7, we can follow Corollary 4 to show the boundedness of e t) for all =,..., N usng: e t) x t) + ê t) )

25 Sensors 26, 6, of 3 Furthermore, n order to obtan the closed-loop system error ultmate bound value for Equaton ) and the no Zeno characterzaton proof, we can follow the same steps hghlghted n Corollares 5 and 6, respectvely. 4. Illustratve Numercal Example In ths secton, the effcacy of the proposed event-trggered decentralzed adaptve control approach s demonstrated n an llustratve numercal example. For ths purpose, we consder the uncertan dynamcal system, whch conssts of fve masses connected serally by sprngs and dampers as depcted n Fgure 2. We use the followng equatons of moton for the -th mass: ẋ t) ẍ t) ẋ t) ẍ t) ẋ 5 t) ẍ 5 t) = = = k m b m x t) ẋ t) k +k ) m b +b ) m k 4 m 5 b 4 m 5 x 5 t) ẋ 5 t) + + m Λ u t) + x t)) + δ 2 x 2 t)) ) Λ u t) + x t)) + δ jx j t)), x t) + ẋ t) m = {2, 3, 4} 2) m 5 Λ 5 u t) + 5 x 5 t)) + δ 54 x 4 t)) 3) where m = Kg, k =.5 N m, b =.4 N sec m, Λ =.7, W o = 3, T, and we set the bass functon as β x t)) = x t). In addton, δ 2 x 2 t)), δ j x j t)) and δ 54 x 4 t)), whch represent the effect of the system nterconnectons, are gven by: δ 2 x 2 t)) = δ j x j t)) = δ 54 x 4 t))) = k k j= k 4 b x 2 t) ẋ 2 t) b j= x j= t) ẋ j= t) b 4 x 4 t) ẋ 4 t) + k j= b j= x j=+ t) ẋ j=+ t), 4) = {2, 3, 4} 5) 6) u t) u 2 t) u 3 t) u 4 t) u 5 t) k k 2 k 3 k 4 m m 2 m 3 m 4 m 5 b b 2 b 3 b 4 x t) x 2 t) x 3 t) x 4 t) x 5 t) Fgure 2. Connected mass-damper-sprng system. The control objectve of each module s to enforce x t) to track a fltered square reference nput c t) under the effect of uncertantes and dsturbances wth reduced communcaton effort by event-trggerng archtecture. For our example, we choose a second-order deal reference model that has a natural frequency of 2 rad/s and a dampng rato of.77 for all S, =,..., 5. In addton, we use a state emulator gan L = 9I 2 and set all ntal condtons to zero for all S, =,..., 5. For the event-trggered decentralzed model reference adaptve control whch s equvalent to L = ), we set Q = I 2 n order to compute P n Equaton ). The condton n Assumpton 4 holds when α j.26 for = {, 5} and α j.3 for = {2, 3, 4}. In ths case, Assumpton 2 s

26 Sensors 26, 6, of 3 satsfed for the couplng terms gven n Equatons 4) 6). For the purpose of event-trggered state emulator-based decentralzed adaptve control, we set R = 3 and Q = I 2 2 n order to compute P n Equaton 48). For l =. and Q = 25I 2, the condton n Assumpton 5 holds when α j 4.2 for = {, 5} and α j 2. for = {2, 3, 4}. In addton, Assumpton 2 s satsfed for couplng terms gven by Equatons 4) 6). For the proposed event-trggered dstrbuted adaptve control, we set Q = I 2 n order to compute P n Equaton ). Note that there are no fundamental stablty condtons for the case of dstrbuted adaptve control. Lastly, for the event-trggerng thresholds, we choose ɛ x =.2 and ɛ u =.2 for = {, 3, 5} and ɛ x =.7 and ɛ u =.7 for = {2, 4}. For the proposed event-trggered decentralzed adaptve control desgn of Theorem and Corollary, Fgures 3 5 represent the results for varous γ and L. In partcular, we frst set γ = 5 and L = n Fgure 3, whch results n a control response wth hgh-frequency oscllatons. In order to suppress these undesred oscllatons, we set L = 9I 2 as seen n Fgure 4. In ths fgure, even though such oscllatons are reduced, the command trackng performance becomes worse as we ncrease L compared to the response n Fgure 3. In addton to ncreasng L, we also ncrease γ n Fgure 5, to mprove command trackng performance wthout causng hgh-frequency oscllatons. In general, f one pcks L to be greater than nne, then t may also be necessary to ncrease γ further to obtan a smlar closed-loop system performance. It should also be mentoned that choosng L and γ to produce both a control response wthout any sgnfcant hgh-frequency oscllatons, and a small unform ultmate bound can be cast as an optmzaton problem, as well. Fgures 6 8 represent the results of the proposed event-trggered dstrbuted adaptve control of Theorem 2 and Corollary 7 for the same γ and L values. Specfcally, we see hgh frequency content n the control sgnal n Fgure 6 when γ = 5 and L =, whch s mtgated by ncreasng the state emulator gan to L = 9I 2, as seen n Fgure 7. In order to enhance the command trackng, whch s degraded by ncreasng the state emulator gan, we ncrease γ as seen n Fgure 8. xt) x2t) x3t) x4t) x5t) r xm x x s r 2 x m2 x 2 x s r 3 x m3 x 3 x s r 4 x m4 x 4 x s r 5 x m5 x 5 x s A) State sgnals ut) u2t) u3t) u4t) u5t) u 2 u s u 3 u s u 4 u s u 5 u s B) Control sgnals u u s Fgure 3. Command followng performance for the proposed event-trggered decentralzed adaptve control approach wth γ = 5 and L =.

27 Sensors 26, 6, of 3 xt) x2t) x3t) x4t) x5t) r xm x x s r 2 x m2 x 2 x s r 3 x m3 x 3 x s r 4 x m4 x 4 x s r 5 x m5 x 5 x s A) State sgnals ut) u2t) u3t) u4t) u5t) u 2 u s u 3 u s u 4 u s u 5 u s B) Control sgnals u u s Fgure 4. Command followng performance for the proposed event-trggered decentralzed adaptve control approach wth γ = 5 and L = 9. xt) x2t) x3t) x4t) x5t) r xm x x s r 2 x m2 x 2 x s r 3 x m3 x 3 x s r 4 x m4 x 4 x s r 5 x m5 x 5 x s A) State sgnals ut) u2t) u3t) u4t) u5t) u 2 u s u 3 u s u 4 u s u 5 u s B) Control sgnals u u s Fgure 5. Command followng performance for the proposed event-trggered decentralzed adaptve control approach wth γ = 2 and L = 9.

28 Sensors 26, 6, of 3 xt) x2t) x3t) x4t) x5t) r xm x x s r 2 x m2 x 2 x s r 3 x m3 x 3 x s r 4 x m4 x 4 x s r 5 x m5 x 5 x s A) State sgnals ut) u2t) u3t) u4t) u5t) u 2 u s u 3 u s u 4 u s u 5 u s B) Control sgnals u u s Fgure 6. Command followng performance for the proposed event-trggered dstrbuted adaptve control approach wth γ = 5 and L =. xt) x2t) x3t) x4t) x5t) r xm x x s r 2 x m2 x 2 x s r 3 x m3 x 3 x s r 4 x m4 x 4 x s r 5 x m5 x 5 x s A) State sgnals ut) u2t) u3t) u4t) u5t) u 2 u s u 3 u s u 4 u s u 5 u s B) Control sgnals u u s Fgure 7. Command followng performance for the proposed event-trggered dstrbuted adaptve control approach wth γ = 5 and L = 9.

29 Sensors 26, 6, of 3 xt) x2t) x3t) x4t) x5t) r xm x x s r 2 x m2 x 2 x s r 3 x m3 x 3 x s r 4 x m4 x 4 x s r 5 x m5 x 5 x s A) State sgnals ut) u2t) u3t) u4t) u5t) u 2 u s u 3 u s u 4 u s u 5 u s B) Control sgnals u u s Fgure 8. Command followng performance for the proposed event-trggered dstrbuted adaptve control approach wth γ = 2 and L = 9. From these results, we observe from the decentralzed adaptve control case that the state emulator-based approach not only gves strngent performance wthout causng hgh frequences n the controller response, but also tolerates the nterconnecton uncertantes of the modules. In addton, the performance of the dstrbuted adaptve controller s better than the decentralzed adaptve controller wth the correspondng desgn parameter settng. The total number of the state and control event trggers of the whole system for the cases n Fgures 3 8 s gven n Fgure 9A,B, respectvely. Fgure 9 shows the drastc decrement of the trggerng number usng the event-trggerng approach and also the further trggerng number decrement due to utlzng the state emulator-based approach. x No event-trggerng γ = 5,L = γ = 5,L =9 γ = 2,L =9 x No event-trggerng γ = 5,L = γ = 5,L =9 γ = 2,L =9 7 7 Number of trggers Number of trggers State sgnal Control sgnal State sgnal Control sgnal A) Decentralzed adaptve control B) Dstrbuted adaptve control Fgure 9. Number of trggers wth respect to the controller desgn parameters. 5. Conclusons The desgn and analyss of event-trggered decentralzed and dstrbuted adaptve control archtectures for uncertan networked large-scale modular systems were presented. For the decentralzed case, t was shown n Secton 2 that the proposed event-trggered adaptve control

30 Sensors 26, 6, of 3 archtecture guarantees system stablty and performance wth no Zeno behavor under some structural condtons stated n Assumptons 4 and 5 that depend on the parameters of the large-scale modular systems and the proposed archtecture. For the dstrbuted case, t was shown n Secton 3 that the proposed event-trggered adaptve control archtecture guarantees the same system stablty and performance wth no Zeno behavor wthout such structural condtons under the assumpton that physcally-nterconnected modules can locally communcate wth each other for exchangng ther state nformaton. In addton to the presented theoretcal fndngs, the effcacy of the proposed event-trggered decentralzed and dstrbuted adaptve control approaches s demonstrated on an llustratve numercal example n Secton 4, where sgnfcant reducton on the overall communcaton cost was obtaned for large-large modular systems n the presence of system uncertantes resultng from modelng and degraded modes of operaton of the modules and ther nterconnectons between each other. For the future work, samplng, data transmsson and computaton delays wll be consdered along wth the proposed results of ths paper, snce they also play an mportant role n the performance of networked control systems. Furthermore, we wll also consder the cases when a set of dagonal elements of the control effectveness matrx s zero and generalze the results of ths paper to cover these so-called loss of control cases. Author Contrbutons: The research documented n ths paper was conducted by Al Albattat, where he receved perodcal support and gudance n theory development and smulatons from Benjamn Gruenwald and Tansel Yucelen on theory development. Conflcts of Interest: The authors declare no conflct of nterest. References. Bullo, F.; Cortes, J.; Martnez, S. Dstrbuted Control of Robotc Networks: A Mathematcal Approach to Moton Coordnaton Algorthms; Prnceton Unversty Press: Prnceton, NJ, USA, Mesbah, M.; Egerstedt, M. Graph Theoretc Methods n Multagent Networks; Prnceton Unversty Press: Prnceton, NJ, USA, Sljak, D.D. Decentralzed Control of Complex Systems; Courer Corporaton: North Chelmsford, MA, USA, Yucelen, T.; Johnson, E.N. Control of mult-vehcle systems n the presence of uncertan dynamcs. Int. J. Control 23, 86, Davson, E.J.; Aghdam, A.G. Decentralzed Control of Large-Scale Systems; Sprnger Publshng Company: New York, NY, USA, Yucelen, T.; Shamma, J.S. Adaptve archtectures for dstrbuted control of modular systems. In Proceedngs of the Amercan Control Conference, Portland, OR, USA, 4 6 June 24; pp Ioannou, P.A.; Sun, J. Robust Adaptve Control; Courer Corporaton: North Chelmsford, MA, USA, Yucelen, T.; Haddad, W.M. A robust adaptve control archtecture for dsturbance rejecton and uncertanty suppresson wth L transent and steady-state performance guarantees. Int. J. Adapt. Control Sgnal Process. 22, 26, Åström, K.J.; Wttenmark, B. Adaptve Control; Courer Corporaton: North Chelmsford, MA, USA, 23.. Lavretsky, E.; Wse, K.A. Robust Adaptve Control. In Robust and Adaptve Control; Sprnger-Verlag London: London, UK, 23.. Narendra, K.S.; Annaswamy, A.M. Stable Adaptve Systems; Courer Corporaton: North Chelmsford, MA, USA, Yucelen, T.; Haddad, W.M. Low-frequency learnng and fast adaptaton n model reference adaptve control. IEEE Trans. Autom. Control 23, 58, Yucelen, T.; de la Torre, G.; Johnson, E.N. Improvng transent performance of adaptve control archtectures usng frequency-lmted system error dynamcs. Int. J. Control 24, 87, Åström, K.J.; Bernhardsson, B. Comparson of Remann and Lebesque samplng for frst order stochastc systems. In Proceedngs of the 4st IEEE Conference on Decson and Control, Las Vegas, NV, USA, 3 December 22; pp Tabuada, P. Event-trggered real-tme schedulng of stablzng control tasks. IEEE Trans. Autom. Control 27, 52,

31 Sensors 26, 6, of 3 6. Heemels, W.; Donkers, M.; Teel, A.R. Perodc event-trggered control for lnear systems. IEEE Trans. Autom. Control 23, 58, Ioannou, P. Decentralzed adaptve control of nterconnected systems. IEEE Trans. Autom. Control 986, 3, Gavel, D.T.; Sljak, D. Decentralzed adaptve control: Structural condtons for stablty. IEEE Trans. Autom. Control 989, 34, Sh, L.; Sngh, S.K. Decentralzed adaptve controller desgn for large-scale systems wth hgher order nterconnectons. IEEE Trans. Autom. Control 992, 37, Spooner, J.T.; Passno, K.M. Decentralzed adaptve control of nonlnear systems usng radal bass neural networks. IEEE Trans. Autom. Control 999, 44, Mrkn, B.M. Decentralzed adaptve controller wth zero resdual trackng errors. In Proceedngs of the 7th Medterranean Conference on Control and Automaton MED99), Hafa, Israel, 28 3 June 999; pp Narendra, K.S.; Oleng, N.O. Exact output trackng n decentralzed adaptve control systems. IEEE Trans. Autom. Control 22, 47, Yucelen, T.; Yang, B.J.; Calse, A.J. Dervatve-free decentralzed adaptve control of large-scale nterconnected uncertan systems. In Proceedngs of the 2 5th IEEE Conference on Decson and Control and European Control Conference, Orlando, FL, USA, 2 5 December 2; pp Mazo, M.; Tabuada, P. Decentralzed event-trggered control over wreless sensor/actuator networks. IEEE Trans. Autom. Control 2, 56, Moln, A.; Hrche, S. Optmal desgn of decentralzed event-trggered controllers for large-scale systems wth contenton-based communcaton. In Proceedngs of the IEEE Conference on Decson and Control and European Control, Orlando, FL, USA, 2 5 December 2; pp Dmarogonas, D.V.; Frazzol, E.; Johansson, K.H. Dstrbuted event-trggered control for mult-agent systems. IEEE Trans. Autom. Control 22, 57, Donkers, M.; Heemels, W. Output-based event-trggered control wth guaranteed-gan and mproved and decentralzed event-trggerng. IEEE Trans. Autom. Control 22, 57, Garca, E.; Cao, Y.; Yu, H.; Antsakls, P.; Casbeer, D. Decentralsed event-trggered cooperatve control wth lmted communcaton. Int. J. Control 23, 86, Fan, Y.; Feng, G.; Wang, Y.; Song, C. Dstrbuted event-trggered control of mult-agent systems wth combnatonal measurements. Automatca 23, 49, Garca, E.; Cao, Y.; Casbeer, D.W. Cooperatve control wth general lnear dynamcs and lmted communcaton: Centralzed and decentralzed event-trggered control strateges. In Proceedngs of the IEEE Amercan Control Conference, Portland, OR, USA, 4 6 June 24; pp Sahoo, A.; Xu, H.; Jagannathan, S. Neural network-based adaptve event-trggered control of nonlnear contnuous-tme systems. In Proceedngs of the IEEE Internatonal Symposum on Intellgent Control, Hyderabad, Inda, 28 3 August 23; pp Sahoo, A.; Xu, H.; Jagannathan, S. Neural network approxmaton-based event-trggered control of uncertan MIMO nonlnear dscrete tme systems. In Proceedngs of the IEEE Amercan Control Conference, Portland, OR, USA, 4 6 June 24; pp Wang, X.; Hovakmyan, N. L adaptve control of event-trggered networked systems. In Proceedngs of the IEEE Amercan Control Conference, Portland, OR, USA, 4 6 June 24; pp Wang, X.; Kharsov, E.; Hovakmyan, N. Real-tme L adaptve control for uncertan networked control systems. IEEE Trans. Autom. Control 25, 6, Albattat, A.; Gruenwald, B.C.; Yucelen, T. Event-trggered adaptve control. In Proceedngs of the ASME Dynamc Systems and Control Conference, Columbus, OH, USA, 28 3 October Albattat, A.; Gruenwald, B.C.; Yucelen, T. Output Feedback Adaptve Control of Uncertan Dynamcal Systems wth Event-Trggerng. In Adaptve Control for Robotc Manpulators; CRC Press/Taylor & Francs Group: Boca Raton, FL, USA, accepted. 37. Lavretsky, E.; Gadent, R.; Gregory, I.M. Predctor-based model reference adaptve control. J. Gud. Control Dyn. 2, 33, Muse, J.A.; Calse, A.J. H adaptve flght control of the generc transport model. In Proceedngs of the AIAA Infotech@Aerospace 2, Atlanta, GA, USA, 2 22 Aprl 2.

32 Sensors 26, 6, of Stepanyan, V.; Krshnakumar, K. MRAC revsted: Guaranteed performance wth reference model modfcaton. In Proceedngs of the IEEE Amercan Control Conference, Baltmore, MD, USA, 3 June 2 July 2; pp Stepanyan, V.; Krshnakumar, K. M-MRAC for nonlnear systems wth bounded dsturbances. In Proceedngs of the IEEE Decson and Control and European Control Conference, Orlando, FL, USA, 2 5 December 2; pp Gbson, T.E.; Annaswamy, A.M.; Lavretsky, E. Improved transent response n adaptve control usng projecton algorthms and closed loop reference models. In Proceedngs of the AIAA Gudance Navgaton and Control Conference, Chcago, IL, USA, 3 August Gbson, T.E.; Annaswamy, A.M.; Lavretsky, E. Adaptve systems wth closed-loop reference models: Stablty, robustness and transent performance. 22, arxv: Godsl, C.; Royle, G. Algebrac Graph Theory; Sprnger: New York, NY, USA, Pomet, J.B.; Praly, L. Adaptve nonlnear regulaton: Estmaton from the Lyapunov equaton. IEEE Trans. Autom. Control 992, 37, Lavretsky, E.; Gbson, T.E.; Annaswamy, A.M. Projecton operator n adaptve systems. 2, arxv: Bernsten, D.S. Matrx Mathematcs: Theory, Facts, and Formulas; Prnceton Unversty Press: Prnceton, NJ, USA, Km, K.; Yucelen, T.; Calse, A.J. A parameter dependent Rccat equaton approach to output feedback adaptve control. In Proceedngs of the AIAA Gudance, Navgaton, and Control Conference, Portland, OR, USA, 8 August Glover, K.; Doyle, J.; Khargonekar, P.P.; Francs, B.A. State space solutons to standard H 2 and H control problems. IEEE Trans. Autom. Control 989, 34, c 26 by the authors; lcensee MDPI, Basel, Swtzerland. Ths artcle s an open access artcle dstrbuted under the terms and condtons of the Creatve Commons Attrbuton CC-BY) lcense