Structural results for partially nested LQG systems over graphs

Transcription

1 Srucural resuls for parially nesed LQG sysems over graphs Ashuosh Nayyar 1 Lauren Lessard 2 American Conrol Conference, pp , 2015 Absrac We idenify a broad class of decenralized oupufeedback LQG sysems for which he opimal conrol sraegies have a simple and inuiive esimaion srucure. We consider cases for which he coupling of dynamics among subsysems and he iner-conroller communicaion are characerized by he same direced graph. For he class of graphs known as mulirees, we show ha each conroller need only esimae he saes of he subsysems i affecs (is descendans) as well as he subsysems i observes (is ancesors). The opimal conrol acion for each conroller is a linear funcion of he esimae i compues and he esimaes compued by is ancesors. Moreover, all sae esimaes may be updaed recursively, much like a Kalman filer. 1 Inroducion The pas decade has seen a resurgence of ineres in decenralized conrol problems; where differen conrollers mus make decisions using differen subses of he available informaion. A cenral quesion in he invesigaion of decenralized problems is wheher he ever-growing informaion hisory available o conrollers be aggregaed wihou compromising performance. In oher words: do he opimal conrollers have convenien sufficien saisics? In his paper, we idenify a broad class of informaion srucures and associaed conrol problems for which he above quesions have an affirmaive answer. We represen informaion srucures by using direced acyclic graphs (DAG), where each node represens boh a subsysem and is associaed conroller, and he edges indicae boh he influence of sae dynamics beween subsysems as well as informaion-sharing beween conrollers. An example is shown in Figure 1. Our main resul, formally saed in Secion 3, may be summarized by he following inuiive saemen: each conroller mus mainain sae esimaes of he subsysems ha i observes (is ancesors) as well as sae esimaes of subsysems ha i affecs hrough is deci- 1 A. Nayyar is wih he Ming Hsieh Deparmen of Elecrical Engineering, Universiy of Souhern California, Los Angeles, CA 90089, USA. ashuosn@usc.edu. A. Nayyar was suppored by NSF under gran number CNS L. Lessard is wih he Deparmen of Mechanical Engineering a he Universiy of California, Berkeley, CA 94720, USA. lessard@berkeley.edu. L. Lessard was suppored by NASA under Gran No. NRA NNX12AM55A Figure 1: Direced acyclic graph (DAG) represening he informaion srucure of a decenralized conrol problem. An edge i j means ha subsysem i affecs subsysem j hrough is dynamics and conroller i shares is informaion wih conroller j (bu no vice versa). sions (is descendans). In he example of Figure 1, he conroller a node 1 mus esimae he saes of nodes {1, 3, 5}, while he conroller a node 4 mus esimae he saes of nodes {2, 4}. The opimal conrol acion for each conroller is a linear funcion of he esimae i compues as well as he esimaes compued by all of is ancesors. In addiion o proving he sufficien saisics, we show ha hey admi a recursive represenaion, similar o ha of he Kalman filer. Our resuls hold for LQG sysems; all dynamics are linear (possibly ime-varying), process noise and measuremen noise is Gaussian, and he cos funcion is quadraic over a finie ime horizon. We mus also impose some resricions on which pairs of nodes may be coupled by he quadraic cos and which pairs of nodes may be driven by correlaed Gaussian disurbances. These assumpions are saed precisely in Secion 2.4. The associaed DAG may be any muliree. In oher words: in he ransiive reducion of he DAG, each pair of nodes can be conneced by a mos one direced pah. For example, if we add he edge 4 5 o Figure 1 he muliree assumpion is now violaed because here are now wo separae direced pahs connecing 2 o 5. A key aspec of his work is ha we consider oupu feedback. While he presence of measuremen noise makes he problem considerably more difficul o solve han he sae-feedback case, we neverheless show ha he opimal conroller has a simple and inuiive srucure. Due o space consrains, we will provide a proof ouline of our main resul only for he specific informaion srucure of Figure 1. Alhough his graph is relaively simple, i capures he salien feaures of our approach and i will be clear how he proof can be generalized o more general graphs. For he ineresed reader, a deailed proof of he general muliree case is available online [16]. 5 1

2 1.1 Prior work The analyical and compuaional difficuly of general decenralized conrol problems wih arbirary informaion srucure has been widely acknowledged [2, 27]. There has been considerable ineres in idenifying classes of informaion srucures ha may be easier o solve. In he eam-heory inspired lieraure on decenralized conrol, a key simplifying feaure is he noion of parial nesedness (PN) [4]. A decenralized LQG problem wih a PN informaion srucure admis a linear conrol sraegy ha is globally opimal. Furher, i can be reduced (a leas for finie horizon problems) o a saic LQG eam problem for which person-by-person opimal sraegies are globally opimal [19]. Despie hese facs, a universal and compuaionally efficien mehodology for finding opimal sraegies for all parially nesed problems remains elusive. Anoher approach is o formulae he conrol problem as a closed-loop norm opimizaion. In his framework, some simplifying properies of he plan and informaion consrain have been idenified. These properies imply convexiy of he se of achievable closed-loop maps. Examples of such properies include quadraic invariance (QI) [20], funnel causaliy [1], and cerain hierarchical archiecures [18]. Despie he convexiy, he opimizaion problem is infinie dimensional in general and herefore hard o solve. A widely used model for PN/QI problems in decenralized conrol is o assume he plan dynamics and he conroller srucure are characerized by he same direced graph. The firs soluions o problems of his kind assumed sae feedback [22, 23]. The case of noisy measuremens (oupu feedback) was found o be considerably more difficul because he problem canno be spli ino separae problems as in he sae feedback case. However, he problem-spliing approach sill holds if only he leaf-nodes have noisy measuremens [5, 17, 24]. The firs soluion o a full oupu feedback case was for a wosubsysem problem [12, 13]. This resul was laer generalized o sar-shaped graphs in [11] and linear chain graphs in [25]. Alhough he above works considered coninuous-ime sysems, he resuls can easily be adaped o a discreeime seing as well. Discree-ime formulaions for decenralized problems were used in he 70 s for reaing parially nesed delayed-sharing cases [7, 21, 26, 28]. More recenly, his noion was generalized o delays characerized in erms of disance on a DAG [8, 9, 10]. The focus of he above works is o find a sae-space represenaion for he opimal decenralized conrol law; here is ypically no immediae way of obaining a meaningful inerpreaion of he opimal conroller saes. One noable excepion is [14], where he conroller saes were inerpreed as minimum mean-squared error (MMSE) esimaes of he plan saes. An alernaive approach is o use a eam-heoreic perspecive. The wo-subsysem oupu feedback problem was solved in his manner in [15]. We use a similar approach in he presen work o exend he oupu feedback resuls o a broader class of DAGs. The advanage of a eam-heoreic approach is ha srucural resuls emerge naurally, and one can deduce he opimal conroller s sufficien saisics wihou solving for gains explicily. Indeed, he paper [15] derives srucural resuls for a finie-horizon formulaion wih a linear ime-varying plan, whereas he works [11, 12, 13, 14, 25] address linear ime-invarian plans and find he infinie-horizon seadysae opimal conroller. The paper is organized as follows. We cover noaion and basic assumpions in Secion 2, he main resuls are presened in Secion 3, and we give a proof ouline in Secions 4 and 5. Finally, we conclude in Secion 6. 2 Preliminaries 2.1 Basic noaion Real vecors and marices are represened by lower- and upper-case leers respecively. Boldface symbols denoe random vecors, and heir non-boldface counerpars denoe paricular realizaions. x T denoes he ranspose of vecor x. E denoes he expecaion operaor. We wrie x = N (µ, Σ) when x is a mulivariae Gaussian random vecor wih mean µ and covariance Σ. We consider discree ime sochasic processes over a finie ime inerval [0, T ]. Time is indicaed using subscrips, and we use he colon noaion o denoe ranges. For example: x 0 T 1 = {x 0, x 1,..., x T 1 }. In general, all symbols are ime-varying. In an effor o presen general resuls while keeping equaions clear and concise, we inroduce a new noaion o represen a family of equaions. For example, when we wrie: x + = Ax + w, we mean ha x +1 = A x + w holds for 0 T 1. Noe ha he subscrip + indicaes ha we incremen o + 1 for he associaed symbol. We similarly overload he summaion symbol by wriing for example x T Qx o mean T 1 x T Q x =0 Whenever is wrien above a binary relaion or below a summaion, i is implied ha 0 T 1. There is no ambiguiy because we use he same ime horizon T hroughou his paper. We denoe subvecors by using superscrips. Subvecors may also be referenced by using a subse of indices as superscrips. For example, for a vecor x = x 1 x 2 x 3, if s = {1, 3}, we wrie: x s = x {1,3} = [ x1 x 3 ] 2

3 When wriing sub-vecors, we will always arrange he componens in increasing order of indices. Given a collecion of random vecors, we will a imes rea he collecion as a concaenaion of vecors arranged in increasing order of node index. For a marix A, le A ij denoes is (i, j) block wih dimensions inferred from he conex. Given wo ses of indices s and r, A s,r is a marix composed of blocks A ij wih i s and j r. 2.2 Graphs Le G(V, E) be a direced acyclic graph (DAG). The nodes are labeled 1 o n, so V = {1,..., n}. If here is an edge from i o j, we wrie (i, j) E. We wrie i j if here is a direced pah from i o j. Tha is, if here exiss a sequence of nodes v 1,..., v m wih v 1 = i and v m = j such ha (v k, v k+1 ) E for all k. By convenion, every node has a direced pah (of lengh zero) o iself. So i is always rue ha i i. We wrie i j if i and j are pah-conneced, ha is, if i j or j i. Oherwise, we say hey are pah-disconneced, and we wrie i j. We can express he pah-connecedness of G using he sparsiy marix, which is he binary marix S {0, 1} n n defined by 1 if j i S ij = 0 oherwise Noe ha differen graphs may have he same sparsiy marix. In general, S is he adjacency marix of he ransiive closure of G. So graphs wih he same ransiive closure also share he same sparsiy marix. By convenion, we assign a opological ordering o he node labels. Tha is, we choose a labeling such ha if j i, hen j i. This is possible for any DAG [3, 22.4]. Therefore, S is always lower-riangular. For example, he sparsiy marix for he graph of Figure 1 is given by S = Given a node i V, we define is ancesors as he se of nodes ha have a direced pah o i. Similarly, we define he descendans as he se of nodes ha i can reach via a direced pah. We use he following noaion for ancesors and descendans respecively. i = {j V j i} i = {j V i j} Ancesors and descendans of i always include i iself. We define he sric ancesors and sric descendans when we mean o exclude i. Specifically, i = i {i} and i = i {i}. We use he noaion i = i i for he se of all nodes ha are pah-conneced o node i. Noe ha i is pariioned as i {i} i. In he graph of Figure 1, for example, 3 = {1, 2, 3}, 3 = {1, 2}, 3 = {3, 5}, 3 = {5}, 3 = {1, 2, 3, 5} Remark 1. Noe ha while i, i, ec. are defined as subses, i is convenien o hink of hem as ordered liss in which he node indices are arranged in increasing order. Thus, in Figure 1, 3 will always be wrien as {1, 2, 3} and no as any oher permuaion of {1, 2, 3}. A node wih no sric descendans is called a leaf node and a node wih no sric ancesors is called a roo node. 2.3 Sysem model The sysem we consider consiss of n subsysems ha may affec one anoher according o he srucure of an underlying DAG, G(V, E). The i h subsysem has sae x i, inpu u i, measuremen y i, process noise w i, and measuremen noise v i. We assume hese are discreeime random processes ha saisfy he following saespace dynamics for all i V. x i + = j i (A ij x j + B ij u j ) + w i y i = j i (C ij x j ) + v i (1) The relaive iming of i h sae, conrol acion and observaion a ime is as shown in Figure 2. Noe ha he observaion y i is generaed afer conrol acion u i is aken. Each of he marices in (1) may be ime-varying, x i u i y i + 1 Figure 2: Relaive iming of sae, acion and observaion a ime. and may even change dimensions wih ime. In an effor o make our noaion more concise, we concaenae he various symbols above and simply wrie x + = Ax + Bu + w y = Cx + v (2) In his condensed noaion, he marices A, B, and C have blocks ha conform o S, he sparsiy marix for G(V, E). In oher words, if S ij = 0 hen A ij = 0, B ij = 0, and C ij = 0. The random vecors in he collecion {x 0, [ w 0 v 0 ],..., [ w T 1 v T 1 ]} (3) 3

4 are referred o as he primiive random variables and are muually independen and joinly Gaussian wih he following known probabiliy densiy funcions x 0 = N (0, Σ ini ) [ w v ] = N (0, [ W U T U V ]) (4) There are n conrollers, one responsible for each of he u i, i = 1,..., n. We define he locally generaed informaion a node i a ime as i i = {y i 0 1, u i 0 1} (5) The informaion available o conroller i a ime is i i = i j = {y j 0 1, uj 0 1 }. (6) j i j i In oher words, each conroller knows he pas measuremens and decisions of is ancesors. So he direced edges of G may be hough of as represening he flow of informaion beween subsysems. Crucially, he same graph G represens boh how he dynamics propagae as well as how informaion is shared in he sysem. The conrollers selec acions according o a conrol sraegy f i = (f0, i f1, i..., ft i 1 ) for i V. Tha is, u i = f i ( i i ) for 0 T 1 (7) Given a conrol sraegy profile f = (f 1, f 2,..., f n ), performance is measured by he finie horizon expeced quadraic cos Jˆ 0 (f) = E f ( [ x T u ] [ Q S S T R ] [x u ] + xt T P final x T ) (8) Expecaion is aken wih respec o he join probabiliy measure on (x 0 T, u 0 T 1 ) induced by he choice of f. I is assumed ha all sysem parameers are universally known. Specifically, Σ ini, P final, as well as he values of A, B, C, Q, R, S, U, V, W for all, are known by all conrollers. 2.4 Assumpions In addiion o he problem specificaions (2) (8), we will make some addiional assumpions abou he underlying DAG and he noise and cos parameers used in (4) and (8) respecively. Firs, we require some definiions. Definiion 1 (muliree). The nodes i, a, b, j V form a diamond if i a j and i b j, and a b. A muliree is a direced acyclic graph ha conains no diamonds. For example, he graph of Figure 1 is a muliree. However, if we add he edge (4, 5), hen he nodes (2, 3, 4, 5) form a diamond and he graph ceases o be a muliree. Definiion 2 (decoupled cos). Define he se X = {Q 0 T 1, R 0 T 1, S 0 T 1, P final } of marices associaed wih he cos funcion. We say ha nodes i, j V have decoupled cos if X ij = 0 for all X X. Definiion 3 (uncorrelaed noise). Define he se Y = {W 0 T 1, V 0 T 1, U 0 T 1, Σ ini } of marices associaed wih he noise and iniial sae saisics. We say ha nodes i, j V have uncorrelaed noise if Y ij = 0 for all Y Y. The noions of decoupled cos and uncorrelaed noise have inuiive inerpreaions. If wo nodes have decoupled cos, hen he insananeous cos a any ime has no cross-erms ha involve boh nodes. If wo nodes have uncorrelaed noise, hen he process and measuremen noises affecing one node are saisically independen of hose affecing he oher. Our assumpions are as follows. (A1) The DAG G(V, E) is a muliree. (A2) For every pair of nodes i, j V, If he pair of nodes has no common ancesor, hen hey have uncorrelaed noise. If he pair of nodes has no common descendans, hen hey have decoupled coss. For he graph of Figure 1, nodes 1 and 4 have neiher a common ancesor nor a common descendan. Therefore, Assumpion A2 would require ha his pair of nodes have boh decoupled cos and uncorrelaed noise. Noe ha because of he muliree assumpion, he only way i and j can have boh a common ancesor and a common descendan is if i j. Assumpion A2 may be expressed in erms of he sparsiy paern S using he following observaions 1. (SS T ) ij = 0 if and only if i j = ; in oher words, if and only if i and j have no common ancesor. 2. (S T S) ij = 0 if and only if i j = ; in oher words, if and only if i and j have no common descendan. So Assumpion A2 may be saed concisely as follows: all marices in X (see Definiion 2) have he same sparsiy as S T S and all marices in Y (see Definiion 3) have he same sparsiy as SS T. For he graph of Figure 1, hese sparsiy paerns are SS T S T S Remark 2. Noe ha Assumpion A2 is more general han assuming ha all cos marices in X and covariance marices in Y are block-diagonal. 4

5 3 Main resuls The problem addressed in his paper is he following. Problem 1 (n-player LQG). For he model (2) (7), and subjec o Assumpions A1 and A2, find a conrol sraegy profile f = (f 1, f 2,..., f n ) ha minimizes he expeced cos (8). The informaion srucure of our problem is parially nesed and herefore, wihou loss of opimaliy, we will resric aenion o linear conrol sraegies [4]. The main resul of his paper is a descripion of sufficien saisics required for an opimal soluion of Problem 1. Theorem 1 (Conrol Resul). In Problem 1, here is no loss in opimaliy in joinly resricing all nodes i V o sraegies of he form where z j = E(x j ij ). u i = K ij zj (9) j i Noe ha z j is he condiional mean of he sae of all nodes ha are pah-conneced o node j based on he informaion available o node j (recall he noaion inroduced in Secion 2.2). Recall ha i i defined in (5) (6) is he informaion available o node i a ime, and his se grows wih ime as more measuremens are observed and more decisions are made. Theorem 1 saes ha conrollers need no remember his enire informaion hisory. Insead, each node j may compue he aggregaed saisic z j, which is an esimae of he curren saes of is ancesors and descendans. The opimal decision a node i, u i, is hen a linear funcion of all such esimaes mainained by he ancesors of node i. Our second resul addresses he evoluion of he esimaes z j. Before we sae his resul, we define he se of marices E i,j. Definiion 4. Consider nodes i and j wih i j. Le i = {k 1, k 2,..., k i } and j = {l 1, l 2,..., l j }. Define a marix E i,j wih i block rows and j block columns as follows: For a = 1, 2,..., i, 1. If k a j, hen he a h block row of E i,j is If k a j and k a = l b, hen he (a, b) block of E i,j is ideniy and he res of a h block row is 0. For example, in Figure 1, we have 3 = {1, 2, 3, 5} and 2 = {2, 3, 4, 5}. Consequenly, E 3,2 I = 0 I I We now sae our second resul. Theorem 2 (Esimaion Resul). If he conrol sraegy is as given in Theorem 1, hen he evoluion of z j described as follows: z j 0 = 0 u j z j + = A j j z j + B j j [ {û ij ] L j (y j C j j z j ) } i j (10) for some marices L j 0 T 1, wih û ij = K ia z a + K ib E b,j z j a j b i j is for i j (11) Remark 3. For linear conrol sraegies described by Theorem 2, he quaniy û ij as defined in (11) is in fac equal o E(u i ij ). The above heorems provide finie dimensional sufficien saisics for all conrollers in he sysem. These resuls should be viewed as srucural resuls of opimal conrol sraegies since hey posulae he exisence of opimal conrollers and esimaors of he form presened above wihou specifying how he marices K ij, L j used in conrol and esimaion can be compued. 4 Proof ouline of Theorem 1 for he sysem in Figure 1 Our proof echnique may be hough of as a sequence of refinemens ha raverses he underlying DAG saring from he leaf nodes and finishing a he roo nodes. Due o space limiaions, we will presen a proof ouline of Theorem 1 as applied o he sysem shown in Figure 1. The mos basic srucural form of conrol sraegies for he sysem illusraed in Figure 1 is simply our iniial = f i ( i i ). We informaion consrain (7), namely, u i refine his srucural form in he following seps: 1. Leaf nodes 4 and 5: We consider a single leaf node (say node 4) and fix arbirary linear conrol sraegies for all nodes excep node 4. We will consider he problem of finding he bes conrol sraegy for node 4 in response o he arbirary choice of linear conrol sraegies of all oher conrollers. This is a cenralized conrol problem for which we can derive a srucural resul by idenifying a suiable sae descripion for he overall sysem as seen by node 4. Wih fixed linear sraegies for all oher nodes, he overall sysem can be viewed as a LQG sysem wih x 1,2,3,4,5, i 1,2,3,5 as he sae vecor and y 2,4, u 2 as he observaion vecor. Because of he srucure of he graph and he sparsiy assumpions on he cos marices, he dynamics and he cos erms associaed wih nodes 1, 3, 5 are decoupled from he dynamics and coss associaed wih node 4. Therefore, for node 4 s cenralized problem, i suffices o consider x 2,4, i 2 as he sae. From 5

6 sandard LQG heory [6], i follows ha node 4 can use a linear conrol law ha is a funcion of is esimae of sae vecor x 2,4, i 2. Tha is, node 4 s linear conrol law is of he form u 4 = g 4 (z 4 ) + h 42 (i 2 ) (12) where z 4 = E(x 2,4 i 2,4 ). A similar argumen for node 5 yields he srucure u 5 = g 5 (z 5 ) + h 5 (i 1,2,3 ) = g 5 (z 5 ) + h 51 (i 1 ) + h 52 (i 2 ) + h (i 3 ) (13) where z 5 = E(x 1,2,3,5 i 1,2,3,5 ). 2. Paren of leaf nodes; node 3: Nex we consider nodes wih only leaf nodes as children. In Figure 1, he only such node is node 3. We fix arbirary linear sraegies for nodes 1 and 2. For node 4, we pick any sraegy ha has he srucural form (12) idenified in Sep 1. We urn our aenion o opimizing sraegies for node 3 and is descendan, node 5. From Sep 1, we know ha u 3 = f 3 (i 1,2,3 ) u 5 = g 5 (z 5 ) + h 5 (i 1,2,3 ) (14) We consider arbirary choices for funcions g0 T 5 1 in (14) and focus on he join opimizaion of f0 T 3 1 and h5 0 T 1. The key hing o noe here is ha boh f 3 and h 5 are funcions of informaion available o node 3. Therefore, we will inroduce a ficiious coordinaor for nodes 3 and 5 ha knows i 1,2,3 and selecs wo decisions: u 3 = f 3 (i 1,2,3 ) ũ = h 5 (i 1,2,3 ) so ha he conrol acion a node 5 can be wrien as u 5 = g 5 (z 5 ) + ũ (15) We now consider he cenralized problem of how his coordinaor should opimally make is decisions. Wih sraegies (or pars of sraegies) of oher conrollers fixed, he overall sysem can be viewed as a LQG sysem from he coordinaor s perspecive wih x 1,2,3,4,5, i 1,2, z 4, z 5 as he sae vecor and y 1,3, u 1 as he observaion vecor. Because of he srucure of he graph and he sparsiy assumpions on he cos marices, he dynamics and he cos erms associaed wih nodes 1,3,5 are decoupled from he dynamics and coss associaed wih node 4. Therefore, for he coordinaor s cenralized problem, i suffices o consider x 1,2,3,5, i 1,2, z 5 as he sae. Using he same argumen as in Sep 1, i suffices ha he coordinaor esimae his new sae. The esimae of z 5 can be wrien as E(z 5 i 1,2,3 ) = E(E(x 1,2,3,5 i 1,2,3,5 ) i 1,2,3 ) (16) By he smoohing propery of condiional expecaion, E(z 5 i 1,2,3 ) = E(x 1,2,3,5 i 1,2,3 ) (17) Therefore, he ficiious coordinaor s opimal sraegy can be wrien as u 3 = g 3 ) + h 31 (i 1 ) + h 32 (i 2 ) ũ = g ) + h 51 (i 1 ) + h 52 (i 2 ) (18) where z 3 = E(x 1,2,3,5 i 1,2,3 ). Combining (15) wih (18) yields he following srucure for u 3 and u 5 u 3 = g 3 ) + h 31 (i 1 ) + h 32 (i 2 ) u 5 = g 5 (z 5 ) + g ) + h 51 (i 1 ) + h 52 (i 2 ) 3. Roo nodes 1 and 2: Finally, we consider he roos nodes 1 and 2. We sar wih node 2. We fix arbirary linear sraegies for node 1 and focus on opimizing sraegies for node 2 and is descendans. Collecing he resuls from Seps 1 and 2, we have u 2 = f 2 (i 2 ) (19a) u 3 = g 3 ) + h 31 (i 1 ) + h 32 (i 2 ) (19b) u 4 = g 4 (z 4 ) + h 42 (i 2 ) (19c) u 5 = g 5 (z 5 ) + g ) + h 51 (i 1 ) + h 52 (i 2 ) (19d) Fix arbirary choices for g 3, g 4, g 5, h 31, h 51, = 0,..., T 1 and focus on he join opimizaion of f 2, h 32, h 42, h 52 for each = 0,..., T 1. Noe ha all he funcions being opimized here are funcions of i 2, he informaion available o node 2. As in Sep 2, inroduce a coordinaor for nodes 2, 3, 4, 5 ha knows i 2 and selecs u 2 = f 2 (i 2 ) ũ 42 = h 42 (i 2 ) ũ 32 = h 32 (i 2 ) ũ 52 = h 52 (i 2 ) (20) Wih sraegies (or pars of sraegies) of oher conrollers fixed, he overall sysem can be viewed as a LQG sysem from he new coordinaor s perspecive wih, i 1, z 3, z 4, z 5 as he sae vecor and y 2 as he observaion vecor. I suffices for he coordinaor o esimae his sae. We now make he following observaion: x 1,2,3,4,5 a) Because of he sparsiy assumpion abou noise saisics (see Assumpion A2), x 1, i 1 are independen of he coordinaor s informaion i 2 a ime. Therefore, E(x 1 i 2 ) = 0 and E(i 1 i 2 ) = 0. b) The coordinaor s esimae of z 4 can be wrien as E(z 4 i 2 ) = E(E(x 2,4 i 2,4 ) i 2 ) = E(x2,4 i 2 ) (21) c) The coordinaor s esimae of z 3 can be wrien as E i 2 ) = E(E(x 1,2,3,5 i 1,2,3 ) i 2 ) = E(x1,2,3,5 i 2 ) (22) 6

7 Furhermore, E(x 1,2,3,5 i 2 ) = E(x 1,2,3,5 E 3,2 x 2,3,4,5 i 2 ) + E(E 3,2 x 2,3,4,5 i 2 ) = E([x ] T i 2 ) + E 3,2 E(x 2,3,4,5 i 2 ) = 0 + E 3,2 E(x 2,3,4,5 i 2 ) (23) d) Similarly, E(z 5 i 2 ) = E 5,2 E(x 2,3,4,5 i 2 ) (24) Based on (21) (24), we observe ha he esimae of he new coordinaor s sae ulimaely only depends on i 2 ). Therefore, (20) can be refined o E(x 2,3,4,5 u 2 = f 2 ) ũ 42 = g 42 ũ 32 = g 32 ) ũ 52 = g 52 ) ) (25) where z 2 = E(x 2,3,4,5 i 2 ). Now merge (25) wih (19) and obain he following srucure for u 2, u 3, u 4, u 5 u 2 = g 2 ) u 3 = g 3 ) + g 32 u 4 = g 4 (z 4 ) + g 42 u 5 = g 5 (z 5 ) + g 52 (26a) ) + h 31 (i 1 ) (26b) ) ) + g (26c) ) + h 51 (i 1 ) (26d) Repeaing a similar argumen for node 1 esablishes he final srucural resul for all nodes. u 1 = g 1 (z 1 ) u 2 = g 2 ) u 4 = g 4 (z 4 ) + g 42 u 3 = g 3 ) + g 31 u 5 = g 5 (z 5 ) + g 51 ) (z 1 ) + g 32 (z 1 ) + g 52 ) ) + g ) (27a) (27b) (27c) (27d) (27e) Observing ha all funcions in (27) are linear and herefore can be wrien in erms of marices yields he resul of Theorem 1 for he sysem of Figure 1. 5 Proof ouline of Theorem 2 The seps 1 3 in he proof of Theorem 1 invoke he cenralized srucural resul from sandard LQG heory [6]; a sufficien saisic for he opimal conroller is he condiional mean of he sae. Bu he cenralized heory also provides a recursive formulaion for he condiional mean (he Kalman filer). Theorem 2 can be proved by augmening he proof of Theorem 1 o include hese esimaor recursions a every sep. We provide an ouline below and refer he reader o [16] for a deailed proof. Consider firs a leaf node, say node 4. This node needs o evaluae z 4 = E(x 2,4 i 2,4 ). The srucure of he graph implies ha he saes of nodes 2 and 4, x 2,4, evolve according o he following dynamics: x 2 + x 4 + = A 22 x 2 + B 22 u 2 + w 2 (28a) = A 42 x 2 + A 44 x 4 + B 42 u 2 + B 44 u 4 + w 4 (28b) y 2 = C22 x 2 + v 2 y 4 = C42 x 2 + C 44 x 4 + v i (28c) (28d) Compuing he esimaor of hese saes based on i 2,4 = {y 2,4 0 1, u2,4 0 1 } is a sandard Kalman esimaion problem wih a recursive updae equaion ha agrees wih he resul of Theorem 2 for j = 4. The esimaes a he oher leaf node, node 5, can be handled in a similar manner. Now focus on node 3 s esimae z 3 = E(x 1,2,3,5 i 1,2,3 ). The difficuly in updaing his esimae is ha he dynamics of x 5 depend on node 5 s conrol acion which node 3 does no fully know. Because of Theorem 1, he acion of node 5 can be wrien as u 5 = K 55 z 5 + K z 3 + K 52 z 2 + K 51 z 5 (29) The las hree erms in (29) are based on informaion available o node 3 and are herefore compuable by node 3. To esimae u 5, herefore, node 3 needs o esimae he firs erm. As we saw in proof of Theorem 1, E(z 5 i 1,2,3 ) = E(x 1,2,3,5 i 1,2,3 ) = z 3 (30) Using he above observaions, we can derive he following updae equaion for z 3 : z 3 + = A 3 3 z 3 + B 3 3 [ u3 û ] L3 (y 3 C 3 3 z 3 ) (31) for some marices L 3 0 T 1, wih û = K 55 z 3 + K z 3 + K 52 z 2 + K 51 z 5. (32) Equaion (32) agrees wih he saemen of Theorem 2 for j = 3. We proceed in a similar manner for he roo nodes 1 and 2. Node 2, for example, needs o updae is esimae z 2 = E(x 2,3,4,5 i 2 ). In order o do so, i needs o esimae he acions of all is descendans. Thus, i needs o esimae he esimaes of nodes 1, 3, 4, 5. Observe ha E(z 1 i 2 ) = 0 because of our assumpions abou noise saisics (Assumpion A2). Finally, as in he proof of Theorem 1, E(z i i 2 ) for i = 3, 4, 5 can be obained as funcions of z 2. Combining hese observaions yields he updae equaion for z 2. 7

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