Distributed Adaptive Fault-Tolerant Control of Nonlinear Uncertain Second-order Multi-agent Systems

Transcription

1 Dtrbuted Adaptve Fault-Tolerant Control of Nonlnear Uncertan Second-order ult-agent Sytem ohen Khall, Xaodong Zhang, Yongcan Cao, aro. Polycarpou, and Thoma Parn Abtract Th paper preent an adaptve fault-tolerant control FTC cheme for a cla of nonlnear uncertan econd-order mult-agent ytem. A local FTC component degned for each agent n the dtrbuted ytem by ung local meaurement and utable nformaton exchanged between neghborng agent. Each local FTC component cont of a fault dagno module and a reconfgurable controller module compred of a baelne controller and two adaptve faulttolerant controller actvated after fault detecton and after fault olaton, repectvely. Under utable aumpton, the cloedloop tablty and leader-follower formaton properte of the dtrbuted ytem are rgorouly etablhed under dfferent operatng mode of the FTC ytem, ncludng the tme-perod before poble fault detecton, between fault detecton and poble olaton, and after fault olaton. I. INTRODUCTION Several modern techncal ytem can be decrbed by mean of dtrbuted mult-agent ytem, that, ytem compred of varou dtrbuted and nterconnected autonomou agent/ubytem. Example of uch ytem nclude cooperatve unmanned vehcle, ntellgent power grd, ar traffc control ytem, etc. In recent year, cooperatve control ung dtrbuted conenu algorthm ha receved gnfcant attenton ee, e.g., 1 and 2. Adaptve method for achevng conenu n uncertan ytem have alo been propoed 3, 4, 5. Snce uch dtrbuted mult-agent ytem need to operate relably at all tme, depte the poble occurrence of faulty behavor n ome agent, the development of fault dagno and accommodaton cheme a crucal tep n achevng relable and afe operaton. In the lat two decade, gnfcant reearch actvte have been conducted n the degn and analy of fault dagno and accommodaton cheme ee, for ntance, 6. ot of thee method utlze a centralzed archtecture, where the dagnotc module degned baed on a global mathematcal model of the overall ytem and requred to have real-tme acce to all enor meaurement. due to lmtaton of computatonal reource and communcaton overhead, uch centralzed method are not utable for large-cale dtrbuted ytem. A a reult, n. Khall wth the Department of Electrcal Engneerng, Wrght State Unverty, Dayton, OH 45435, USA khall.4@wrght.edu. X. Zhang wth the Department of Electrcal Engneerng, Wrght State Unverty, Dayton, OH 45435, USA xaodong.zhang@wrght.edu. Y. Cao wth Department of Electrcal and Computer Engneerng, Unverty of Texa, San Antono, TX 78249, USA yongcan.cao@uta.edu.. Polycarpou wth the KIOS Reearch Center for Intellgent Sytem and Network, Department of Electrcal and Computer Engneerng, Unverty of Cypru, Ncoa, Cypru mpolycar@ucy.ac.cy. T. Parn wth Imperal College London and Unverty of Trete t.parn@gmal.com. recent year, there ha been gnfcant reearch nteret n dtrbuted fault dagno and accommodaton cheme ee, for ntance, 7, 8, 9, 1. Th paper preent a method for detectng, olatng, and accommodatng fault n a cla of dtrbuted nonlnear uncertan mult-agent ytem. A fault-tolerant control component degned for each agent n the dtrbuted ytem by utlzng local meaurement and utable nformaton exchanged between neghborng agent. Each local FTC component cont of two man module: 1 the onlne health montorng fault dagno module cont of a bank of nonlnear adaptve etmator. One of them the fault detecton etmator, whle the ret are fault olaton etmator; and 2 the controller fault accommodaton module cont of a baelne controller and two adaptve faulttolerant controller employed after fault detecton and after fault olaton, repectvely. Under utable aumpton, the cloed-loop tablty and leader-followng formaton properte are etablhed for the baelne controller and adaptve fault-tolerant controller, repectvely. In prevou paper, a centralzed FDI and fault-tolerant control cheme preented n 11, and a dtrbuted FDI and fault-tolerant control cheme for frt-order mult-agent ytem preented n 12. Th paper extend the reult n thee paper by generalzng the fault-tolerant control method to the cae of leader-follower formaton of dtrbuted econd-order multagent ytem. II. GRAPH THEORY NOTATIONS A drected graph G a par V, E, where V = {υ 1,,υ m } a et of node, E V V a et of edge, and m the number of node. An edge an ordered par of dtnct node υ j,υ meanng that th node can receve nformaton from jth node. For an edge υ j,υ, node υ j called the parent node, node υ the chld node, and υ j a neghbor of υ. An undrected graph can be condered a a pecal cae of a drected graph where υ,υ j E mple υ j,υ E for any υ,υ j V. The et of neghbor of node υ denoted by N = { j : υ j,υ E }. The weghted adjacency matrx A = a j R m m aocated wth the drected graph G defned by a =, a j > f υ j,υ E, and a j = otherwe. The topology of an nterconnecton graph G ad to be fxed, f each node ha a fxed neghbor et and a j fxed. It clear that for undrected graph a j = a j. The Laplacan matrx L = ι j R m m defned a ι = j N a j and ι j = a j, j. Both A and L are ymmetrc for undrected graph, and L potve emdefnte.

2 III. PROBLE FORULATION A. Dtrbuted ult-agent Sytem odel Conder a et of nterconnected agent wth the dynamc of the th agent, = 1,,, beng decrbed by the followng econd-order dynamc ṗ = v 1 v = φ x + u + η x,t + β t T f x, p where x = R v 2, u R, are the tate vector and nput vector of the th agent, repectvely. Addtonally, φ : R 2 R, η : R 2 R + R, f : R 2 R are mooth vector feld. The model gven by 1 ẋ = + x φ x + u 1 2 repreent the known nomnal dynamc of the th agent wth φ beng the known nonlnearty, whle the healthy ytem decrbed by ẋ = φ x + 1 u x + + η 1 x,t. 3 The dfference between the nomnal model 2 and the actual healthy ytem dynamc 3 due to the vector feld η repreentng the modelng uncertanty n the tate dynamc of the th agent. The term β t T f x denote the change n the dynamc of th agent due to the occurrence of a fault. Specfcally, β t T repreent the tme profle of a fault whch occur at ome unknown tme T, and f x an unknown nonlnear fault functon. In th paper, the tme profle functon β aumed to be a tep functon.e., β t T = f t < T, and β t T = 1 f t T whch denote an abrupt fault. The ytem model 1 allow the occurrence of fault n multple agent but t aumed there only a ngle fault n each agent at any tme. Remark 1: The dtrbuted mult-agent ytem model gven by 1 a nonlnear generalzaton of the double ntegrator dynamc condered n lterature for ntance, 2. In th paper, n order to nvetgate the fault-tolerance and robutne properte, the fault functon β f x and modelng uncertanty η are ncluded n the ytem model. For olaton purpoe, we aume that there are r type of poble nonlnear fault functon n the fault cla aocated wth the th agent; pecfcally, f x belong to a fnte et of functon gven by F = { f 1 x,, f r x }. 4 Each fault functon f, = 1,,r, decrbed by f x = θ T g x, 5 where θ, for = 1,,, an unknown parameter vector aumed to belong to a known compact et Θ.e., θ Θ R q, and g : R 2 R q a known mooth vector feld. A decrbed n 11, the fault model decrbed by 4 and 5 characterze a general cla of nonlnear fault where the vector feld g repreent the functonal tructure of the th fault affectng the tate equaton, whle the unknown parameter vector θ characterze the fault magntude. The objectve of th paper to develop a robut dtrbuted fault dagno and fault-tolerant leader-followng formaton control cheme for a cla of dtrbuted multagent ytem decrbed by 1. The followng aumpton are made throughout the paper: Aumpton 1. Each modelng uncertanty, repreented by η x,t n 1, ha a known upper bound,.e., η x,t η x,t, x R 2, 6 where the the boundng functon η known and unformly bounded. Aumpton 2. The communcaton topology among follower undrected, and the leader ha drected path to all follower. Aumpton 1 characterze the cla of modelng uncertanty under conderaton. The bound on the modelng uncertanty needed n order to dtnguh between the effect of fault and modelng uncertanty durng the fault dagno proce 13. Aumpton 2 needed to enure that the nformaton exchange among agent uffcent for the team to acheve the dered team goal. B. Fault-Tolerant Control Structure In th paper, we nvetgate the FTC problem of leaderfollowng formaton. Specfcally, the objectve to develop dtrbuted robut FTC algorthm to guarantee that each agent output converge to a gven predefned formaton around a tme-varyng leader even n the preence of modelng uncertanty and fault. Fg. 1: Dtrbuted FTC archtecture for the th agent The dtrbuted FTC archtecture hown n Fgure 1. Frt of all, we defne three mportant tme ntant: T the fault occurrence tme; T d > T the tme ntant when a fault detected; T ol > T d the tme ntant when the montorng ytem pobly provde a fault olaton decon, that, whch fault n the cla F ha actually occurred. The tructure of the fault-tolerant controller for the th agent take on the followng general form: 11 ω = u = b ω,x,x J,x,t, for t < T d b D ω,x,x J,x,t, for T d t < T ol b I ω,x,x J,x,t, for t T ol h ω,x,x J,x,t, for t < T d h D ω,x,x J,x,t, for T d t < T ol h I ω,x,x J,x,t, for t T ol 7

3 where ω the tate vector of the dtrbuted controller, x the tme-varyng bounded leader tate, x J contan the tate varable of neghborng agent that drectly communcate wth agent,.e., J = { j : j N }; b, b D, b I and h, h D, h I are nonlnear functon to be degned accordng to the followng qualtatve objectve: 1 In a fault free mode of operaton, a baelne controller guarantee the output of th agent x t hould track the formaton around a leader tme-varyng output x, even n the poble preence of plant modelng uncertanty. 2 If a fault detected, the baelne controller reconfgured to compenate for the effect of the yet unknown fault, that, the fault-tolerant controller degned n uch a way to explot the nformaton that a fault ha occurred, o that the controller may recover ome control performance. Th new controller hould guarantee the boundedne of ytem gnal and ome leader-followng formaton performance, even n the preence of the fault. 3 If the fault olated, then the controller reconfgured agan. The econd fault-tolerant controller degned ung the nformaton about the type of fault that ha been olated o a to mprove the control performance. IV. BASELINE CONTROLLER DESIGN In th ecton, we degn the baelne controller and nvetgate the cloed-loop ytem tablty and performance before fault occurrence. Wthout lo of generalty, let the leader be agent number wth a reference output.e., x t = p t v t T where p = v. The baelne controller for the th agent degned a follow: u = j N k j αp p p j + p j + γv v j φ x η + κgn Ξ, 8 where p and p j are the contant dered dtance between the leader and agent and j, repectvely, κ a potve bound on v.e., κ v, gn the gn functon, Ξ = j N k j ɛp p p j + p j +ρv v j, N the et of neghborng agent that drectly communcate wth the th agent ncludng the leader a agent number wth p =, k j are potve contant for j N, and α, γ, ρ, and ɛ are potve contant to be defned n Lemma 1. Notce that k l =, for l / N. Frt, we need the followng Lemma: Lemma 1: Conder a potve defnte quare matrx Ψ R. Defne I A = ρψ ɛψ, P =, 9 αψ γψ ɛψ ρψ where I the dentty matrx, ρ > ɛ, and ρ,ɛ,γ,α >. The matrx Q = PA + A T P negatve defnte f the followng condton are met: ɛ γɛ = αρ, αɛ + ργ < µ ρ 2 mn, 4αρ 2 ɛ 2 < µ mn, 1 where µ mn the mallet egenvalue of Ψ. Proof: Ung 9, the matrx Q can be obtaned a 2αɛΨ Q = 2 ρψ ɛγ + αρψ 2 ρψ ɛγ + αρψ 2 2ɛΨ 2γρΨ The egenvalue are found ung the followng charactertc equaton: I Q = I + 2αɛΨ 2 ρψ + ɛγ + αρψ 2 ρψ + ɛγ + αρψ 2 I 2ɛΨ + 2γρΨ 2 =. Note that  ˆB Ĉ ˆD =  ˆD Ĉ ˆB f  and Ĉ commute. Alo, t can be hown that 2 I h 1 Ψ h 2 Ψ = =1 2 h 1 µ h 2 µ, where h 1 and h 2 are polynomal, and µ the th egenvalue of Ψ. Thu, we have I Q = = ɛµ + αɛ + ργµ 2 + 4αɛµ 2 ργµ 2 ɛµ ρµ γɛ + αρµ 2 2. To have all the egenvalue n the left-half complex plane, the coeffcent of and the contant need to be potve. Snce µ >, the followng condton guarantee that the egenvalue of Q le n the left-half complex plane: { ɛ + αɛ + ργµ > γɛ αρ 2 µ 2 + 4αɛ 2 + 2ρ γɛ + αρ µ ρ 2 >. The above nequalte are guaranteed by the condton gven n 1. Thu, the proof completed. The followng reult characterze the cloed-loop tablty and leader-followng formaton performance properte of the overall mult-agent ytem pror to any fault occurrence. Theorem 1: In the abence of fault n the th agent, ung the baelne controller decrbed by 8, the leader-follower formaton control acheved aymptotcally wth a tmevaryng reference tate,.e. p t p t p and v t v t a t. Proof: Baed on 8 and 1, the cloed-loop ytem dynamc, n the abence of a fault.e., for t < T, are gven by ṗ = v v = k j α p j + γṽ j + η η + κgn Ξ, 12 j N where p j = p p p j p j and ṽ j = v v j. We repreent the collectve cloed-loop dynamc a x = A x + ζ ζ, 13 1 v where A defned n Lemma 1 wth the table matrx Ψ = L + dag{k 1,k 2,,k } 14, L the communcaton graph Laplacan matrx, x = p T ṽ T T R 2 n whch p the column tack vector of p = p p p and ṽ the column tack vector of ṽ = v v, the term ζ R and ζ R are defned a ζ ζ = η 1 η T 14 = ζ1 ζ, 15 where ζ = η + κ gn Ξ, = 1,,. We conder the

4 followng Lyapunov functon canddate: V = x T P x, 16 where P a potve defnte matrx defned n Lemma 1. Then, the tme dervatve of the Lyapunov functon 16 along the oluton of 13 gven by V = x T Q x + 2 x T P ζ ζ 1 v, 17 where Q defned n Lemma 1. Baed on 9 and 14, we have x T ζ P = ɛ p T Ψζ + ρṽ T Ψζ = =1 j N k j ɛ p j + ρṽ j η. 18 By ung the ame reaonng logc for the other term n 17 and ubttutng them nto 17, we have V = x T Q x =1 =1 j N k j ɛ p j + ρṽ j η v j N k j ɛ p j + ρṽ j η + κgn Ξ. 19 Baed on Aumpton 1, we have η v j N k j ɛ p j + ρṽ j η + κ j N k j ɛ p j + ρṽ j gn Ξ. 2 Therefore, by applyng the above nequalty to 19, we obtan V x T Q x. Ung Lemma 1, we know that V negatve defnte, and p and ṽ converge to zero a t. Therefore, the leader-followng formaton control reached aymptotcally,.e., p t p t p and v t v t a t. Remark 2: The baelne controller guarantee the convergence of the leader-followng conenu algorthm n the abence of fault. The analy an extenon of the conenu algorthm gven n 14 by conderng the preence of modelng uncertanty η and by ung more control parameter e.g., ρ and α to allow certan flexblty n controller degn. V. DISTRIBUTED FAULT DETECTION AND ISOLATION The dtrbuted fault detecton and olaton FDI archtecture compred of local FDI component, wth one FDI component degned for each of the agent. The objectve of each local FDI component to detect and olate fault n the correpondng agent. Specfcally, each local FDI component cont of a fault detecton etmator FDE and a bank of r nonlnear adaptve fault olaton etmator FIE, where r the number of dfferent nonlnear fault type n the fault et F 4 aocated wth the correpondng agent. Under normal condton, each local FDE montor the correpondng local agent to detect the occurrence of any fault. If a fault detected n a partcular agent, then the correpondng r local FIE are actvated for the purpoe of determnng the partcular type of fault that ha occurred n the agent. The FDI degn for each agent follow the generalzed oberver cheme archtectural framework 6. A. Dtrbuted Fault Detecton Baed on the agent model decrbed by 1, the FDE for each agent choen a: ˆx = Λ 1 φ x ˆx + x + x 1 + u, 21 where ˆx R 2 denote the etmated local tate, Λ = λ p λv R 2 2 a potve defnte etmator gan matrx. For each local FDE, let ε = x ˆx = ε p ε v T denote the tate etmaton error of the th agent. Then, before fault occurrence.e., for t < T, by ung 1 and 21, the etmaton error dynamc are gven by ε = Λ ε η x,t Therefore, ung 22, we have ε v ν, where ν t = t e λ v t τ η x,τdτ + x e λ v t, 23 x a conervatve bound on the ntal tate etmaton error.e., ε v x. Note that the ntegral term n the above threhold can be ealy mplemented a the output of a lnear flter wth the nput gven by η x,t. Thu, we have the followng: Fault Detecton Decon Scheme: The decon on the occurrence of a fault detecton n the th agent made when the modulu of the etmaton error.e., ε v generated by the local FDE exceed t correpondng threhold.e., ε v > ν. B. Dtrbuted Fault Iolaton Now, aume that a fault detected n the th agent at ome tme T d ; accordngly, at t = T d the FIE n the local FDI component degned for the th agent are actvated. Each local FIE degned baed on the functonal tructure of a partcular fault type n the agent ee 5. Specfcally, the followng r nonlnear adaptve etmator are degned a olaton etmator for the th agent: for = 1,,r, ˆx = Λ x ˆx 1 + x φ + x 1 + u + ˆθ T g x 24, where ˆθ, for = 1,,, and = 1,,r, the etmate of the fault parameter vector n the th agent, and Λ = λ p λv a dagonal potve defnte matrx. The adaptaton n the olaton etmator due to the unknown fault parameter vector θ. The adaptve law for updatng each ˆθ derved by ung the Lyapunov ynthe approach 15, wth the projecton operator P retrctng ˆθ to the correpondng known et Θ. Specfcally, f we let ε t = x ˆx = ε p εv T be the etmaton error generated by the th FIE aocated wth the th agent, then the followng

5 adaptve algorthm choen: ˆθ = P Θ {γ g x ε v }, where γ > a contant learnng rate. Baed on 1 and 24, the tate etmaton error dynamc n the preence of fault gven by ε = Λ ε η + x 1,t θ T g x, where ε the tate etmaton error, and θ = ˆθ θ the parameter etmaton error. Therefore, by ung the trangle equalty, a bound on the tate etmaton error can be obtaned a εv ς t, where ς t e λ v t τ η + ξ T d g x dτ + x e λ v t T d, where x a pobly conervatve bound on the ntal tate etmaton error.e., εv T d x, and ξ repreent the maxmum fault parameter vector etmaton error,.e., θ ˆθ t ξ. The form of ξ depend on the geometrc properte of the compact et Θ 11. For ntance, aume that the parameter et Θ a hyperphere or the mallet hyperphere contanng the et of all poble ˆθ t wth center O and radu R ; then we have ξ = R + ˆθ t O. The fault olaton decon cheme baed on the followng ntutve prncple: f fault occur at ome tme T and detected at tme T d, then a et of threhold functon ς t can be degned uch that the etmaton error generated by the th etmator atfe εv t ς t for all t T d. In the fault olaton procedure, f for a partcular fault olaton etmator b, the etmaton error atfe εv b t > ς b t for ome fnte tme t > T d, then the poblty of the occurrence of correpondng fault type can be excluded. Baed on th ntutve dea, the followng fault olaton decon cheme deved. Dtrbuted fault olaton decon cheme: If for each b {1,,r }\{}, there ext ome fnte tme t b > T d, uch that εv b t b > ς btb, then the occurrence of fault n the th ubytem concluded. VI. FAULT-TOLERANT CONTROLLERS In th ecton, the degn and analy of the fault-tolerant control cheme are rgorouly nvetgated for two dfferent operatng mode of the cloed-loop ytem: 1 durng the perod after fault detecton and before olaton, and 2 after fault olaton. A. Accommodaton before Fault Iolaton After the fault detected at tme t = T d n agent, the olaton etmator decrbed n Secton V.B are actvated to determne the partcular type of fault that ha occurred. eanwhle, the nomnal controller reconfgured to enure the ytem tablty and ome trackng performance after fault detecton. In the followng, we decrbe the degn of the fault-tolerant controller ung adaptve trackng technque. Before the fault olated, no nformaton about the fault functon avalable. Adaptve approxmator uch a neuralnetwork model can be ued to etmate the unknown fault functon β f. The term adaptve approxmator 16 ued to repreent nonlnear multvarable approxmaton model wth adjutable parameter, uch a neural network, fuzzy logc network, polynomal, plne functon, etc. Specfcally, we conder lnearly parametrzed network e.g., radalba-functon network wth fxed center and varance decrbed a follow: ˆf x, ˆϑ = ϱ j=1 c j ϕ j x, 25 where ϕ j repreent the fxed ba functon, and ˆϑ = colc j : j = 1,,ϱ the adjutable weght of the nonlnear approxmator. In the preence of a fault, ˆf provde the adaptve tructure for the onlne approxmaton of the unknown fault functon f x. Th acheved by adaptng the weght vector ˆϑ t. Remark 3. The objectve of adaptve parameter etmaton n the FDI procedure and the fault accommodaton procedure are dfferent. The goal of adaptve parameter etmaton n the cae of FDI learnng,.e., to approxmate the fault functon ee for example the fault olaton etmaton model gven by 24, whle the objectve durng fault accommodaton to modfy the feedback control law va parameter adaptaton o a to tablze the ytem and guarantee ome trackng performance n the preence of a fault. However, the parameter do not necearly converge to the true parameter unle the condton of pertence of exctaton aumed. In th paper, we do not aume the pertence of exctaton condton. Therefore, the ytem dynamc decrbed by 1 can be rewrtten a 1 φ ẋ = x + x 1 + u +η x,t + ˆf x,ϑ + δ x 26, where δ = f x ˆf x,ϑ the network approxmaton error for the th agent, and ϑ the optmal weght vector gven by { } ϑ = arg nf up f x ˆf x, ˆϑ, ˆϑ Θ x X where X R 2 denote the et to whch the varable x belong for all poble mode of the controlled ytem. To mplfy the ubequent analy, n the followng we aume that the boundng condton on the network approxmaton error are global, o we et X = R 2. For each network, we make the followng aumpton on the network approxmaton error: Aumpton 3. for each = 1,,, δ α δ x, 27 where δ a known potve boundng functon, and α an unknown contant. Baed on the ytem model 26, the neural network model 25, and Aumpton 3, an adaptve neural controller can be degned ung neural-network-baed approxmaton and adaptve boundng control technque 16. Specfcally, we

6 conder the followng controller algorthm: u = φ x k j α p j + γṽ j ψ j N ˆf x, ˆϑ t η + κgn Ξ 28 ˆϑ = Γ k j ɛ p j + ρṽ j ϕ x 29 j N ψ = ˆα δ x gn k j ɛ p j + ρṽ j 3 j N ˆα = ϒ k j ɛ p j + ρṽ j δ x, 31 j N where ˆϑ an etmaton of the neural network parameter vector ϑ, ϕ = col ϕ j : j = 1,,ϱ the collectve vector of fxed ba functon, ˆα an etmaton of the unknown contant α, and Γ and ϒ are ymmetrc potve defnte learnng rate matrce. We can repreent the collectve cloed-loop dynamc a x = A x + ζ ζ, 32 1 v + f where A gven n Lemma 1, and x = p T ṽ T T defned n a mlar way a n 13, the term ζ R and ζ R are defned n 14 and 15, the term f R defned a f = f 1 + δ 1 ψ 1 f + δ ψ T, 33 where f = ϑ T ϕ, and ϑ = ϑ ˆϑ the network parameter etmaton error aocated wth the th agent. To derve the adaptve algorthm and to nvetgate analytcally the tablty properte of the cloed-loop ytem, we conder the followng Lyapunov functon canddate: V = x T P x + ϑ T Γ 1 ϑ + α T ϒ 1 α, 34 where P defned n Lemma 1, ϑ = ϑ T 1 ϑ T T the collectve parameter etmaton error, α = α 1 α T the collectve boundng parameter etmaton error defned a α = α ˆα, and Γ = dag{γ 1,,Γ } and ϒ = dag{ϒ 1,,ϒ } are contant learnng rate matrce. Followng the ame procedure a gven n the proof of Theorem 1, ung 33, t can be hown that the tme dervatve of the Lyapunov functon 34 along the oluton of 32 atfe V = x T Q x + 2 =1 k j ɛ p j + ρṽ j η v j N k j ɛ p j + ρṽ j η + κgn Ξ j N + ϑ T j N k j ɛ p j + ρṽ j ϕ Γ 1 ˆϑ + j N k j ɛ p j + ρṽ j δ ψ α ϒ 1 ˆα. Therefore, by ung 2 and electng the adaptve algorthm for ˆϑ a 29, we have V x T Q x + 2 =1 j N k j ɛ p j + ρṽ j δ ψ α ϒ 1 ˆα. 35 By ung 3, and baed on Aumpton 3, we have k j ɛ p j + ρṽ j δ ψ = Ξ δ ˆα δ gnξ j N =1 Ξ α δ, 36 where Ξ defned n 8. By ung 35 and 36, we have V x T Q x + 2 k j ɛ p j + ρṽ j α δ j N α ϒ 1 ˆα. Therefore, by ung 31, we have V x T Q x, 37 where Q gven n Lemma 1. Thu, we conclude that x, ˆϑ, and ˆα are unformly bounded. By ntegratng both de of 37, t can be ealy hown that x L 2. Snce x L L 2 and x L, baed on Barbalat Lemma 15, we can conclude that the leader-followng formaton between agent output reached aymptotcally,.e., x a t. The aforementoned degn and analy procedure ummarzed n the followng theorem: Theorem 2: Suppoe that the boundng Aumpton 3 hold. Then, f a fault detected, the adaptve faulttolerant law 28, the weght parameter adaptve law 29, and the boundng parameter adaptve law 3 and 31 guarantee that all the gnal and parameter etmate are unformly bounded,.e., x, ˆϑ, and ˆα are bounded, and leader-followng formaton acheved aymptotcally wth a tme-varyng reference tate,.e. p t p t p and v t v t a t. B. Accommodaton after Fault Iolaton In th ecton, we decrbe and analyze the adaptve faulttolerant controller employed after fault olaton. Let u now aume that the olaton procedure decrbed n Secton V.B provde the nformaton that fault ha been olated at tme T ol. Then, for t T ol, ung 5 the dynamc of the ytem take on the followng form: ẋ = 1 x + 1 φ x + u + η + θ T g x. 38 The control objectve to have the output x, = 1,,, track the tme-varyng output of the leader and form a formaton around the leader. After the olaton of the fault type,.e., t T ol, the followng adaptve fault-tolerant controller adopted: u = φ x k j α p j + γṽ j ˆθ T g x j N η + κgn Ξ 39

7 ˆθ = Γ j N k j ɛ p j + ρṽ j g x, 4 where ˆθ an etmaton of the unknown fault parameter vector, and Γ a ymmetrc potve defnte learnng rate matrx. Then, we have the followng: Theorem 3: Aume that fault occur at tme T and that t olated at tme T ol. Then, the fault-tolerant controller 39 and fault parameter adaptve law 4 guarantee that all tate are bounded, and the leader-followng formaton acheved aymptotcally wth a tme-varyng reference tate,.e. p t p t p and v t v t a t ; Proof: Baed on 38 and 39, the cloed-loop ytem dynamc are gven by ṗ = v v = k j α p j + γṽ j + η η + κgn Ξ j N + θ T g x. We can repreent the collectve cloed-loop dynamc a x = A x + ζ ζ 1 v + f 41 where A gven n Lemma 1, and x = p T ṽ T T defned n a mlar way a n 13, the term ζ and ζ are defned n 14 and 15, and f R defned a f = f 1 f T 42 where f = θ T g, and θ = θ ˆθ the parameter etmaton error correpondng to the th agent. We conder the followng Lyapunov functon canddate: V = x T P x + θ T Γ 1 θ, 43 where P defned n Lemma 1, θ = θ 1 T θ T the collectve parameter etmaton error, and Γ = dag{γ 1,,Γ } a potve defnte learnng rate matrx. Then, ung 18 and a mlar reaonng logc for 42, the tme dervatve of the Lyapunov functon 43 along the oluton of 41 gven by V = x T Q x + 2 k j ɛ p j + ρṽ j η v =1 j N k j ɛ p j + ρṽ j η + κgn Ξ j N + θ T j N k j ɛ p j + ρṽ j g Γ 1 ˆθ, where Q defned n Lemma 1. Therefore, ung 2 and choong the adaptve law a 4, we have V x T Q x. Then, the proof can be concluded by ung a mlar reaonng logc a reported n the proof of Theorem 2. VII. SIULATION RESULTS In th ecton, a mulaton example of a networked mult-agent ytem contng of 5 agent condered to llutrate the effectvene of the dtrbuted fault-tolerant T control method. The dynamc of each agent gven by 1 φ ẋ = x + x 1 + u + η + β t T f x, where, for = 1,,5, x = p v T the tate of the th agent, and u the nput of th agent. The nomnal term n the dynamc of each agent φ x = v 2. The unknown modelng uncertanty n the local dynamc of the agent are aumed to be a nuodal gnal η =.5nt whch aumed to be bounded by η =.6. The objectve have each agent follow a vrtual leader x gven v by ẋ = wth zero ntal condton and alo keep.5 nt a formaton around the leader wth p 1 = 4, p 2 = 2, p 3 =, p 4 = 2, p 5 = 4. The Laplacan matrx of the nterconnecton graph of agent gven a L = The vrtual leader only communcate wth the econd agent.e., k 2 = 1. The matrx Ψ = L + dag{,1,,,} ha the mnmum egenvalue of µ mn =.13, and α = 3, γ = 3, ɛ =.1, and ρ = 1 atfy the condton gven n Lemma 1. The fault cla under conderaton defned a 1 A proce fault functon f 1 = θ 1g1, where g1 = p condered a the frt fault type, and the magntude of th fault condered a θ A proce fault functon f 2 = θ 2g2, where g2 = p np condered a the econd fault type, and the magntude of the fault condered a θ 2 1. The etmator gan for the fault detecton etmator choen a λ = 2. For fault olaton etmator, λ = 2 ha been choen. A radal ba functon RBF neural network ued for approxmaton of the fault after t detecton and before t olaton. The RBF network condered n th paper cont of 21 neuron wth 21 adjutable gan parameter. The center of radal ba functon are equally dtrbuted on nterval 1, 1 wth a varance of.5. The ntal parameter vector of the neural network et to zero. We et the learnng rate a Γ = 5 and conder an unknown contant bound on the network approxmaton error,.e., δ = 1. The learnng rate choen a ϒ =.1. After fault olaton, the neural-network-baed adaptve fault-tolerant controller reconfgured to accommodate the pecfc fault that ha been olated. We et the learnng rate Γ =.2 wth a zero ntal condton ee 4. Fgure 2 and Fgure 3 how the fault detecton and olaton reult when the frt proce fault cla.e., f1 1 = θ 1 1g1 1 wth a magntude of.8 occur to agent 1 at T = 3 econd. A can be een from Fgure 2, the redual correpondng to the output generated by the local FDE degned for agent 1 exceed t threhold mmedately after fault occurrence. Therefore, the proce fault n agent 1 tmely detected. It can be een n Fgure 3 that the redual correpondng to the FIE aocated wth the frt fault type alway reman

8 below the threhold, whle the redual correpondng to the FIE aocated wth the econd fault type exceed the threhold at approxmately t = 3.6 econd. Thu, baed on the fault olaton decon cheme decrbed n Secton V.B the occurrence of fault type 1 can be concluded. Regardng the performance of the adaptve fault-tolerant controller, a can be een from Fgure 4, the leaderfollowng formaton acheved ung the propoed adaptve FTC even after fault occurrence, whle the agent cannot acheve the leader-followng formaton and become untable wthout the FTC controller ee Fgure 5. Fg. 4: The trackng error n the cae of a proce fault n agent 1: wth adaptve fault-tolerant controller Fg. 2: The cae of a proce fault n agent 1: fault detecton redual old and blue lne and the correpondng threhold dahed and green lne generated by the local FDE VIII. CONCLUSION In th paper, we nvetgate the problem of a dtrbuted FDI and FTC for a cla of mult-agent uncertan econdorder ytem. By ung on-lne dagnotc nformaton, adaptve FTC controller are developed to acheve the leaderfollowng formaton wth a tme-varyng leader n the preence of fault. The cloed-loop tablty and leader-followng Fg. 5: The trackng error n the cae of a proce fault n agent 1: wthout adaptve fault-tolerant controller formaton properte are rgorouly etablhed under dfferent mode of the FTC ytem. The extenon to ytem wth more general tructure an nteretng topc for future reearch. REFERENCES Fg. 3: The cae of a proce fault n agent 1: the fault olaton redual old and blue lne and the correpondng threhold dahed and green lne generated by the two FIE of agent 1 1 Z. Qu, Cooperatve control of dynamcal ytem: applcaton to autonomou vehcle. New York: Sprnger-Verlag, W. Ren and R. Beard, Dtrbuted Conenu n ult-vehcle Cooperatve Control: Theory and Applcaton. London, U.K.: Sprnger- Verlag, L. Cheng, Z. Hou,. Tan, Y. Ln, and W. Zhang, Neural-networkbaed adaptve leader-followng control for mult-agent ytem wth uncertante, IEEE Tranacton on Neural Network, vol. 21, pp , Augut A. Da and F. L. Lew, Dtrbuted adaptve control for ynchronzaton of unknown nonlnear networked ytem, Automatca, vol. 46, no. 12, pp , Y. Hu, H. Su, and J. Lam, Adaptve conenu wth a vrtual leader of multple agent governed by locally lpchtz nonlnearty,

9 Internatonal Journal of Robut and Nonlnear Control, vol. 23, no. 9, pp , Blanke,. Knnaert, J. Lunze, and. Staroweck, Dagno and Fault-Tolerant Control. Berln: Sprnger, F. Arrchello, A. arno, and F. Perr, Oberver-baed decentralzed fault detecton and olaton trategy for networked multrobot ytem, IEEE Tranacton on Control Sytem Technology, DOI: 1.119/TCST R. Ferrar, T. Parn, and.. Polycarpou, Dtrbuted fault detecton and olaton of large-cale dcrete-tme nonlnear ytem: An adaptve approxmaton approach, IEEE Tranacton on Automatc Control, vol. 57, no. 2, pp , I. Shame, A.. Texera, H. Sandberg, and K. H. Johanon, Dtrbuted fault detecton for nterconnected econd-order ytem, Automatca, vol. 47, pp , X. Yan and C. Edward, Robut decentralzed actuator fault detecton and etmaton for large-cale ytem ung a ldng-mode oberver, Internatonal Journal of Control, vol. 81, no. 4, pp , X. Zhang, T. Parn, and.. Polycarpou, Adaptve fault-tolerant control of nonlnear ytem: a dagnotc nformaton-baed approach, IEEE Tranacton on Automatc Control, vol. 49, pp , Augut Khall, X. Zhang,.. Polycarpou, T. Parn, and Y. Cao, Dtrbuted adaptve fault-tolerant control of uncertan mult-agent ytem, n 9th IFAC Sympoum on Fault Detecton, Supervon and Safety for Techncal Procee, September A. Emam-Naen,.. Akhter, and S.. Rock, Effect of model uncertanty on falure detecton: the threhold elector, IEEE Tranacton on Automatc Control, vol. 33, pp , Y. Cao and W. Ren, Dtrbuted coordnated trackng wth reduced nteracton va a varable tructure approach, IEEE Tranacton on Automatc Control, vol. 57, pp , January P. A. Ioannou and J. Sun, Robut Adaptve Control. Englewood Clff, NJ: Prentce Hall, J. Farrell and.. Polycarpou, Adaptve Approxmaton Baed Control. Hoboken, NJ: J. Wley, 26.