Adaptive Architectures for Distributed Control of Modular Systems

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1 Adaptve Archtectures for Dstrbuted Control of Modular Systems ansel Yucelen and Jeff S. Shamma Abstract We consder the dstrbuted control of large-scale modular systems,.e., systems consst of physcally nterconnected and possbly heterogeneous submodules that use local nformaton to acheve a gven set of global objectves. A graphtheoretc approach s used to model the unknown physcal nteractons between submodules and utlze an adaptve control algorthm to learn and cancel the effect of such nteractons as well as the submodule-level modelng uncertantes asymptotcally by allowng submodules to locally communcate wth each other through the graph. he key feature of our framework s to bound the msmatch error between the actual and desred closed-loop modular system performance by a constant that s a pror known and does nether depend on the underlyng graph topology nor uncertantes. I. INRODUCION Large-scale systems consst of nteractng components that exchange matter, energy, or nformaton and emerge frequently n engneerng such as network systems, power systems, communcaton systems, process control systems, water systems, hghway systems, and ar traffc control systems see ] and references theren. An mportant class of large-scale systems s modular systems n whch there exsts a physcal nterconnecton between submodules. A major challenge n the control of modular systems s assocated wth unknown physcal nteractons between submodules and submodule-level modelng uncertantes. Although fxed-gan robust control desgn approaches can be used n ths case, they requre the knowledge of uncertanty bounds for the constructon of stablzng laws. Characterzaton of these bounds s not trval due to practcal constrants, because t requres extensve and costly verfcaton and valdaton procedures. Furthermore, n the face of system faults, structural damage, or reconfguraton, the actual bounds may change and exceed ther estmates. In such cases, the performance of a modular system may be poor and t s desrable for a dstrbuted control system to be able to learn and cancel these uncertantes. o that end, the control framework of ths paper bulds on adaptve control theory. he dstrbuted adaptve control concept s orgnally proposed by Ioannou ] and snce that date numerous contrbutons were made to the feld wth notable ones ncludng 3 ]. Specfcally, Refs. 6 consder the tradtonal approach, where no communcaton s allowed between dfferent submodules and controllers. As a result, subsystems follow a. Yucelen s wth the Department of Mechancal and Aerospace Engneerng, Mssour Unversty of Scence and echnology, Rolla, MO 649-, USA e-mal: yucelen@mst.edu. J. S. Shamma s wth the School of Electrcal and Computer Engneerng, Georga Insttute of echnology, Atlanta, GA 333-, USA e-mal: shamma@gatech.edu. hs research was supported by the Unversty of Mssour Research Board and the Offce of Naval Research under grant N Fg.. A modular system represented by a crcle graph wth a local communcaton between controllers and b local communcaton between submodules. hck lnes denote physcal nterconnectons represented by the graph, thn lnes denote communcatons, and S and C denote submodule and controller, =,..., N, respectvely. Fg.. Expanded vew of a submodule n Fg. b x and u denote the state vector and control sgnal, respectvely, of submodule, and j x j and α j x j denote the unknown physcal nteracton effects and the nformaton exchange, respectvely, between submodules and j. gven desred closed-loop trajectory wth bounded error. In order to overcome ths drawback and show that the msmatch error between the actual and desred closed-loop system performance vanshes asymptotcally, Refs. 7 requres that every local controller needs to access the desred closed-loop trajectores of all submodules. hs type of communcaton topology mposes an assumpton on the choce of desgn parameters,.e., they have to satsfy a condton dependng on the norm of unknown weghts appearng n the uncertanty parameterzaton of unknown physcal nteractons see, for example, 8 n Ref. 7, 6 n Ref. 8, or 3 n Ref.. Although authors of Refs. 8 and try to relax ths condton by employng addtonal adaptaton algorthms, t may not be possble that all desred closed-loop trajectores s avalable to all submodules. here are also contrbutons, where communcaton between each submodule s allowed n the desgn see, for example, ] for a desgn n the context of fault accommodaton. For the dstrbuted control of large-scale modular systems consst of heterogeneous submodules, ths paper consders a graph-theoretc approach to model the unknown physcal nteractons between submodules. Specfcally, we utlze a novel adaptve control algorthm to learn and cancel the effect of such nteractons as well as the submodule-level possbly exstng modelng uncertantes asymptotcally by allowng submodules to locally communcate wth each other through the graph see Fg. b and ts expanded vew n Fg.. he key feature of our framework s to bound the

2 msmatch error between the actual and desred closed-loop modular system performance by a constant that s a pror known and does nether depend on the underlyng graph topology nor uncertantes. II. NOAIONS AND DEFINIIONS In ths paper, 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 + resp., R + denotes the set of postve resp., nonnegatve-defnte real numbers, R n n + resp., R n n + denotes the set of n n postve-defnte resp., nonnegatve-defnte real matrces, S n n denotes the set of n n symmetrc real matrces, and D n n denotes the n n real matrces wth dagonal scalar entres. In addton, we wrte λ mn A resp., λ max A for the mnmum resp., maxmum egenvalue of the Hermtan matrx A, daga for the dagonal matrx wth the vector a on ts dagonal, for the transpose operator, for the nverse operator, tr for the trace operator, for the Eucldan norm, for the nfnty norm, F for the Frobenus matrx norm, and for the equalty by defnton. Furthermore, for a sgnal zt = z t, z t,..., z n t] R n, t R +, the truncated L norm and the L norm ] are defned as, respectvely z τ t L max n sup t τ z t and zt L max n sup t z t. We adopt graphs 3] to encode physcal nteractons and communcatons between submodules,.e., f there exsts a physcal nteracton between a set of submodules represented on a graph, then there also exsts a communcaton between them represented on the same graph as n Fg. b. In partcular, a graph G s defned by a set 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 n j. he degree of a node s gven by the number of ts neghbors. Lettng d be the degree of node, DG dagd, d = d,..., d N ], represents the degree matrx of graph G. A path... L s a fnte sequence of nodes such that k k, k =,..., L, and a graph G s connected f and only f there s a path between any par of dstnct nodes. III. PROBLEM FORMULAION We consder a modular system M consstng of N uncertan heterogeneous submodules defned on the graph G such that the physcal nteractons between submodules are unknown. In partcular, the dynamcs of submodule, V G, s gven by S : ẋ t = A x t + B u t + D j j xj t +δ x t ], x = x, If DG NI N, then we say that the control problem s weakly dstrbuted snce there may exst at least one node such that d = N. Smlarly, we say that the control problem s strctly dstrbuted f d N for all V G,.e., DG < NI N. hroughout ths paper, we assume that all submodules formng a modular system are physcally connected wthout loss of generalty, and therefore, a graph encodng ths nteracton s connected. where x t R n s the submodule state vector, u t R m s the submodule control nput restrcted to the class of admssble controls consstng of measurable functons, j : R nj R m s an unknown physcal nteracton wth respect to submodule j, j V G, such that, j E G, δ : R n R m s a submodule uncertanty, A R n n s the submodule system matrx, B R n m s an unknown submodule control nput matrx, D R n m s the submodule uncertanty nput matrx, and the par A, B s controllable. Assumpton 3.. he unknown physcal nteracton between submodules and j,, j E G, and the unknown submodule uncertanty are parameterzed as j x j = F j α j xj, xj R nj, δ x = G β x, x R n, 3 respectvely, where F j R f m and G R g m are unknown weght matrces and α : R nj R f and β : R n R g are the correspondng bass functons. Remark 3.. For the case when the bass functons α j xj t and β x t n Assumpton 3. are unknown, the parameterzatons n and 3 can be relaxed by resortng to unversal approxmaton tools and restrctng the evoluton of all submodule state vectors to a compact subset of R n, V G. For example, neural networks are used for ths purpose n Refs. 4 and. Assumpton 3.3. he unknown submodule control nput matrx s parameterzed as B = D Λ, 4 where Λ R m m + D m m s an unknown control effectveness matrx. Next, consder the reference system, V G, capturng a desred closed-loop performance for submodule, V G, gven by S r : ẋ r t = A r x r t + B r c t, x r = x r, where x r t R n s the reference state vector, c t R m s a gven bounded command, A r R n n s the reference system matrx, and B r R n m s the command nput matrx. Assumpton 3.4. here exsts K R m n and K R m m such that A r = A +B K and B r = B K hold wth A r beng Hurwtz. he statement of the dstrbuted control problem s now gven as follows. Consder the modular system M on the graph G consstng of submodules S, V G, subject to Assumptons 3., 3.3, and 3.4. Let u t R m be a dstrbuted controller C that accesses the state x t of S and receves the bass α j xj t only from the local neghborng submodules j,, j E G. Our am s to desgn a local C for each submodule to asymptotcally drve the msmatch between each S and S r to zero n the presence of unknown physcal nteractons and submodule-level uncertantes; and acheve a guaranteed performance n terms of boundng

3 the msmatch between each S and S r by a pror known constant for all tme that nether depend on the underlyng graph topology nor uncertantes, subject to the assumpton that the ntal msmatch error s less than ths known constant. In what follows, we ntroduce the graph-theoretc dstrbuted adaptve control framework for modular systems to address Secton IV, provde the novel learnng rate adjustment approach to acheve a guaranteed performance to address Secton V, numercally llustrate the proposed dstrbuted control algorthm Secton VI, and summarze conclusons Secton VII. IV. DISRIBUED ADAPIVE CONROL hs secton ntroduces the graph-theoretc dstrbuted adaptve control framework. For ths purpose, consder the modular system M on the graph G consstng of submodules S, V G. Usng 4, can be rewrtten as where ẋ t = A r x t + B r c t + D Λ ũ t, 6 ũ t u t + Λ j F j α j xj t +Λ G β x t K x t K c t. 7 Snce the uncertan terms n 7 can be parameterzed by usng the unknown weghts,.e., Λ F j, Λ G, K, K, 8 j and ther correspondng bass functons,.e., α j xj t, β x t, x t, c t, 9 j we defne W σ Λ j F j α j xj t +Λ G β x t K x t K c t, for notatonal convenence, where W conssts of these unknown weghts and σ x t, x j t, c t,, j E G, conssts of the correspondng bass functons. Next, let the dstrbuted adaptve feedback controller of S, V G, be gven by C : u t Ŵ tσ x t, x j t, c t, where Ŵt s an estmate of W satsfyng the update law Ŵ t γ σ x t, x j t, c t x t x r t P D, Ŵ = Ŵ, wth γ R + beng a fxed learnng rate a state-dependent learnng rate adjustment approach s ntroduced n the next secton and P R n n + S n n s a soluton of the Lyapunov equaton = A rp + P A r + R, 3 where R R n n + S n n. Now, lettng e t x t x r t, 4 W t Ŵt W, submodule-level closed-loop dynamcs are gven by ė t A r e t D Λ W tσ x t, x j t, c t, e = e, 6 W t γ σ x t, x j t, c t e tp D, W = W. 7 he next theorem presents the frst result of ths secton. heorem 4.. Consder the modular system M on the graph G consstng of submodules S, V G, descrbed by, subject to Assumptons 3., 3.3, and 3.4. Consder, n addton, the reference system gven by and the dstrbuted adaptve feedback controller gven by and for submodule, V G. hen, the soluton e t, W t gven by 6 and 7 s Lyapunov stable for all e, W and t R + and lm t e t = for all, V G. Proof. o show Lyapunov stablty of the soluton e t, W t gven by 6 and 7 for all, V G, consder the Lyapunov functon canddate V e, W = e P e + γ tr W Λ W Λ. Note that V, = and V e, W > for all e, W,. Dfferentatng ths Lyapunov functon canddate along the trajectores of 6 and 7 yelds V e t, W t = e tr e t. Now, t follows from V N = V e, W that V = N = e tr e t, and hence, the soluton e t, W t s Lyapunov stable for all e, W and t R+ and for all, V G. o show that lm t e t = holds for all, V G, note that σ x t, x j t, c t s bounded for t R +. It follows from 6 that ė t s bounded, and hence, V s bounded for t R +. As a drect consequence of Barbalat s lemma ] lm t V =, whch shows that lm t e t = holds for all, V G. heorem 4. hghlghts the steady-state performance guarantees of the proposed dstrbuted adaptve control archtecture gven by and for the modular system M. he next theorem presentng the second result of ths secton establshes the transent performance propertes of ths archtecture. heorem 4.. Consder the modular system M on the graph G consstng of submodules S, V G, descrbed by, subject to Assumptons 3., 3.3, and 3.4. Consder, n addton, the reference system gven by and the dstrbuted adaptve feedback controller gven by and for submodule, V G. hen, for t R +, e t L and λ max P e + γ λ mn P et L W Λ F ], 8 N max, V G e t L, 9

4 where et e t,..., e N t ]. Proof. Snce V e t, W t for t R + heorem 4., ths mples that V e, W V e, W. Usng V e, W λmax P e + γ W Λ F and V e, W λ mn P e t, t follows from V e, W V e, W that e t λ max P e + γ λ mn P W Λ F ]. Snce and ths bound s unform, λ max P e + γ e τ t L W ] Λ F. λ mn P holds, and hence, 8 s a drect consequence due to the fact that ths expresson holds unformly n τ. Fnally, snce N et L e t L, 3 = 9 s now mmedate. Remark 4.3. Although heorem 4. hghlghts the transent performance propertes of the proposed dstrbuted adaptve control archtecture, computaton of the gven L bound may not be practcally possble snce such bounds depends not only to the known desgn parameters but also to the uncertan terms W and Λ, whch may not be avalable a pror. V. PERFORMANCE-ORIENED APPROACH O DISRIBUED ADAPIVE CONROL hs secton presents a state-dependent rate adjustment approach for the dstrbuted adaptve control framework ntroduced n the prevous secton to acheve a pror known transent tme worst-case guaranteed modular system performance. We start wth the followng defnton. Defnton.. Let z R p be a real column vector and z H z Hz, 4 be a weghted Eucldan norm, where H R p p + S p p. We say φ z H, φ : R p R, s a restrcted potental functon defned on D ɛ { z H : z H, ɛ }, f the followng statements hold: If z H =, then φ z H =. If z H D ɛ and z H, then φ z H >. If z H ɛ, then φ z H. v φ z H s contnuously dfferentable on Dɛ. v If z H D ɛ, then φ d z H >, where dφ z H φ d z H d z. 6 H Remark.. A canddate restrcted potental functon satsfyng the condtons of Defnton. has the form φ z H = whch has the dervatve φ d z H = z H ɛ z H, z H D ɛ, 7 ɛ z H ɛ z H wth respect to z H. Next, let the dstrbuted adaptve feedback controller of S, V G, be gven by, where Ŵt s an estmate of W satsfyng the update law Ŵ t γ φ d e P σ x t, x j t, c t e tp D, Ŵ = Ŵ, 8 wth γ φ d e P beng a state-dependent learnng rate, where γ R + acts as a scalng factor, and P R+ n n S n n s a soluton of the Lyapunov equaton gven by 3, where R R n n + S n n. Now, submodule-level closedloop dynamcs are gven by 6 and W t γ φ d e P σ x t, x j t, c t e tp D, W = W. 9 he next theorem presents the man result of ths secton that not only contrbutes to the modular systems lterature but also contrbutes to the adaptve control lterature. heorem.3. Consder the modular system M on the graph G consstng of submodules S, V G, descrbed by, subject to Assumptons 3., 3.3, and 3.4. Consder, n addton, the reference system gven by and the dstrbuted adaptve feedback controller gven by and 8 for submodule, V G. If e P < ɛ, then the soluton e t, W t gven by 6 and 9 s Lyapunov stable for all admssble e, W and t R+ and lm t e t = for all, V G, wth the guaranteed a pror known transent performance bound gven by e t P < ɛ, t R +. 3 for all, V G. Proof. Defne V : D e R dm W R + as V e, W = φ e P +γ tr W Λ W Λ, 3 where D e { e t : e t P < ɛ }, 3 and dm W denotes the dmenson of W. For any ξ >, let Ω { e t, W t D e R dm W : V e, W ξ }, 33 denote the level sets of V e, W. Note that V, = and V e, W > for all admssble e, W,. Furthermore, consderng dφ e t P = φ d e t P e dt tp ė t, 34 t follows from 3 that V e t, W t = φ d e t P e tr e t, 3

5 for e t, W t D e R dm W and t R +. Next, we follow a smlar approach to 7, 8] for concludng our proof. he level sets of V e, W, Ω, are compact and nvarant. In partcular, the set Ω for ξ > s closed by the contnuty of V e, W for e t, W t D e R dm W. Let Q be the set of all ponts n Ω such that V e t, W t =,.e., Q { e t, W t D e R dm W : V e t, W t = } = { e t, W t D e R dm W : e = }. 36 It now follows that all solutons approach the largest nvarant set R n Q. hs concludes the proofs snce R s composed of all ponts n whch e t =, 37 D Λ W tσ x t, x j t, c t =, 38 for all, V G. Remark.4. Note that followng the formulaton gven below n the proof of heorem 4., 3 can be stated as P e t L < ɛ. 39 Furthermore, usng 3 along wth, we can wrte et L < Nɛ. 4 Comparng 8 wth 39 respectvely, 9 wth 4, one can conclude that 39 respectvely, 4 presents a guaranteed transent performance bound ɛ respectvely, Nɛ, whch s a pror known and nether depend on the underlyng graph topology nor uncertantes, subject to the assumpton that e P < ɛ. hs assumpton can be easly satsfed, for example, by lettng x = x r, V G and hence, e =, V G, snce the submodule state vectors are accessble 9]. VI. ILLUSRAIVE NUMERICAL EXAMPLE We consder the modular system M on the lne graph G consstng of 8 coupled nverted pendulums as shown n Fg. 3. he pendulum S has a local control nput C, whch has access to the angular poston θ t, the angular velocty θ t, and to the neghborng state nformaton. he pendulum couplngs enter through a shared sprng on one sde and a shared damper on the other. We use the followng t t 3 t 8 t Fg Coupled nverted pendulums. equatons of moton for the th pendulum lnearzed about ts uprght equlbrum state ] ] ẋ t = x.. t + Λ u t ] ] + x t, 4 ] ] ẋ k t = x.k.k k t + Λ k u k t ] ] ] ] + xk+t + xk t, k =, 3,..., 7, 4 ] ] ẋ 8 t = x 8 t + Λ 8 u 8 t ] ] + x7 t, 43 where x t θ t, θ t], Λ =. for = {, 3,, 7} and Λ = for = {, 4, 6, 8}. Note that 4, 4, and 43 can be equvalently wrtten n the form gven by and satsfy Assumptons 3., 3.3, and 3.4. We assume that there s no a pror knowledge exsts about the parameters of these coupled nverted pendulums. For our smulatons, we choose the reference system gven by ] ] ẋ r t = x 4 8 r t + c 4 t, 44 correspondng to a natural frequency ω n = rad/sec and a dampng rato of ζ =. Furthermore, we choose R = I, use local commands, local states, and neghborng states for the bass functons, and set all ntal condtons to zero. Here, our am s to track a gven snusodal reference command c t = sn.t, V G. For the proposed dstrbuted adaptve control desgn of heorem.3, Fgs. 4 and present the results. Specfcally, Fg. 4 shows the command followng and control hstores and Fg. shows the learnng rate adjustment and error hstores of the 8 coupled nverted pendulums. In these fgures, we use the restrcted potental functon of Remark. wth ɛ = to acheve a strngent command followng performance. VII. CONCLUSION o contrbute prevous studes of dstrbuted control of modular systems, we presented a graph-theoretc learnng approach to asymptotcally learn and cancel the effect of unknown physcal nteractons between submodules as well as submodule-level uncertantes. he key feature of our framework was to bound the msmatch error between the actual and desred closed-loop modular system performance by a constant that s a pror known for the dstrbuted control desgn. Future research wll nclude studes for not only achevng strngent performance specfcatons but also lmtng the frequency content of the control sgnal 8] for robustness guarantees aganst hgh-frequency dynamcal system content such as unmodeled dynamcs.

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