Hierarchical Design of Decentralized Receding Horizon Controllers for Decoupled Systems

Transcription

1 Hierarchical Design of Decenralized Receding Horizon Conrollers for Decoupled Sysems Tamás Keviczy, Francesco Borrelli, Gary J. Balas* Absrac We consider a se of decoupled dynamical sysems and an opimal conrol problem where cos funcion and consrains couple he dynamical behavior of he sysems. The coupling is described hrough a conneced graph where each sysem is a node and, cos and consrains of he opimizaion problem associaed o each node are only funcion of is sae and he saes of is neighbors. In a recen repor [1] we have proposed a mehod for designing decenralized receding horizon conrollers (RHC). Each RHC conroller is associaed o a differen node and compues he local conrol inpus based only on he saes of he node and of is neighbors. For such a decenralized scheme, sabiliy and feasibiliy can be ensured in differen ways, by modifying cos, consrains and communicaions srucure. In his paper we focus on decenralized RHC conrol design hrough hierarchical decomposiion. A cerain prioriy is assigned o each node of he graph and nodes wih high prioriies compue conrol laws for nodes wih lower prioriies. We sudy how o ensure sabiliy and feasibiliy of such scheme when he explici feasibiliy domains of he decenralized RHC are available. Moreover, we propose a hierarchical RHC conrol scheme wih sabiliy and feasibiliy guaranees. I. INTRODUCTION The ineres in decenralized conrol goes bac o he sevenies. Wang and Davison were probably he firs in [2] o envision he increasing ineres in decenralized conrol sysems when conrol heory is applied o solve problems for large scale sysems. Since hen he ineres has grown more han exponenially despie some non-encouraging resuls on he complexiy of he problem [3]. Decenralized conrol echniques oday can be found in a broad specrum of applicaions ranging from roboics and formaion fligh o civil engineering. Such a wide ineres maes a survey of all he approaches ha have appeared in he lieraure very difficul and goes also beyond he scope of his paper. Approaches o decenralized conrol design differ from each oher in he assumpions hey mae on: (i) he ind of ineracion beween differen sysems or differen componens of he same sysem (dynamics, consrains, objecive), (ii) he model of he sysem (linear, nonlinear, consrained, coninuous-ime, discree-ime), (iii) he model of informaion exchange beween he sysems, and (iv) he conrol design echnique used. Dynamically coupled sysems have been he mos sudied. In [2] he auhors consider a linear ime-invarian sysem and give sufficien condiions for he exisence *Deparmen of Aerospace Engineering and Mechanics, Universiy of Minnesoa, 107 Aerman Hall, 110 Union Sree S.E., Minneapolis, MN 55455, {eviczy,borrelli,balas}@aem.umn.edu of feedbac laws which depend only on parial sysem oupus. Recenly, in [4] he auhors inroduce he concep of quadraic invariance of a consrain se wih respec o a sysem. The problem of consrucing decenralized conrol sysems is formulaed as one of minimizing he closed loop norm of a feedbac sysem subjec o consrains on he conrol srucure. The auhors show ha quadraic invariance is a necessary and sufficien condiion for he exisence of decenralized conrollers. In [5] he auhors consider spaially inerconneced sysems, i.e. sysems composed of idenical linear ime-invarian sysems which have a srucured inerconnecion opology. By exploiing he inerconnecion opology, he auhors sudy decenralized analysis and sysem conrol design using l 2 -induced norms and LMIs. In his paper we focus on decoupled sysems. The problem of decenralized conrol for decoupled sysems can be formulaed as follows. A dynamical sysem is composed of (or can be decomposed ino) disinc dynamical subsysems ha can be independenly acuaed. The subsysems are dynamically decoupled bu have common objecives and consrains which mae hem inerac beween each oher. Typically he ineracion is local, i.e. he goal and he consrains of a subsysem are funcion of only a subse of oher subsysems saes. The ineracion will be represened by an ineracion graph, where he nodes represen he subsysems and an arc beween wo nodes denoes a coupling erm in he goal and/or in he consrains associaed o he nodes. Also, ypically i is assumed ha he exchange of informaion has a special srucure, i.e., i is assumed ha each subsysem can sense and/or exchange informaion wih only a subse of oher subsysems. Ofen he ineracion graph and he informaion exchange graph coincide. A decenralized conrol scheme consiss of disinc conrollers, one for each subsysem, where he inpus o each subsysem are compued only based on local informaion, i.e., on he saes of he subsysem and is neighbors. Our ineres in decenralized conrol for dynamically decoupled sysems arises from he abundance of newors of independenly acuaed sysems and he necessiy of avoiding cenralized design when his becomes compuaionally prohibiive. Newors of vehicles in formaion, producion unis in a power plan, cameras a an airpor, mechanical acuaors for deforming surface are jus a few examples. Each newor has is peculiariy. In formaion fligh for insance he coupling consrains arise from collision avoidance. The ineracion graph is full (each vehicle has o avoid all he

2 oher vehicles) bu i is ofen approximaed wih a imevarying graph based on a closes spaial neighbors model. In a recen paper [1] we have proposed a mehod for designing decenralized receding horizon conrollers (RHC). A cenralized RHC conroller is broen ino disinc RHC conrollers of smaller sizes. Each RHC conroller is associaed o a differen node and compues he local conrol inpus based only on he saes of he node and of is neighbors. The main issue regarding decenralized schemes is ha he inpus compued locally are, in general, no guaraneed o be globally feasible and o sabilize he overall eam. In general, sabiliy and feasibiliy of decenralized schemes are very difficul o prove and/or oo conservaive. As in classical RHC design sabiliy and feasibiliy can be ensured in differen ways, by modifying cos and consrains. In decenralized RHC he communicaions srucure is anoher degree of freedom which can be used for such goal. Enforcing hierarchy in he communicaion lins can be one way o exploi his degree of freedom. In his repor we follow he lines of he hierarchical decomposiion approaches which have been proposed in [6], [7], [8]. We focus on hierarchical decomposiion of an RHC scheme. We assume ha each node of he graph has a cerain prioriy and use a sraegy where nodes wih higher prioriies compue conrol laws for nodes wih lower prioriies. The focus of he paper is o sudy he feasibiliy and sabiliy of such schemes when explici feasibiliy domains of he decenralized RHC are available. II. PROBLEM FORMULATION Consider a se of N v linear decoupled dynamical sysems, he i-h sysem being described by he discree-ime imeinvarian sae equaion: x i +1 = f i (x i, u i ) (1) where x i Rni, u i Rmi, f i : R ni R mi R ni are sae, inpu and sae updae funcion of he i-sysem, respecively. Le X i R ni and U i R mi denoe he se of feasible saes and inpus of he i-h sysem, respecively: x i X i, u i U i, 0 (2) where X i and U i are given polyopes. We will refer o he se of N v consrained sysems as a eam sysem. Le x R Nv ni and ũ R Nv mi be he vecors which collec he saes and inpus of he eam sysem a ime, i.e. x = [x 1,..., xnv ], ũ = [u 1,..., unv ], wih x +1 = f( x, ũ ) (3) We denoe by (x i e, u i e) he equilibrium pair of he i-h sysem and ( x e,ũ e ) he corresponding equilibrium for he eam sysem. So far he sysems belonging o he eam sysem are compleely decoupled. We consider an opimal conrol problem for he eam sysem where cos funcion and consrains couple he dynamic behavior of individual sysems. We use a graph opology o represen he coupling in he following way. We associae he i-h sysem o he i-h node of he graph, and if an edge (i, j) connecing he i-h and j-h node is presen, hen he cos and he consrains of he opimal conrol problem will have a componen which is a funcion of boh x i and x j. The graph will be undireced, i.e. (i, j) A (j, i) A. Before defining he opimal conrol problem, we need o define a graph G = {V, A} (4) where V is he se of nodes V = {1,..., N v } and A V V he ses of arcs (i, j) wih i V, j V. Once he graph srucure has been fixed, he opimizaion problem is formulaed as follows. Denoe wih x i he saes of all neighboring sysems of he i-h sysem, i.e. x i = {x j R nj (j, i) A}, x i Rñi wih ñ i = j (j,i) A nj. Analogously, ũ i R mi denoes he inpus o all he neighboring sysems of he i-h sysem. Le g i,j (x i, x j ) 0 (5) define he inerconnecion consrains beween he i-h and he j-h sysems, wih g i : R ni R nj R nci,j. We will ofen use he following shorer form of he inerconnecion consrains defined beween he i-h sysem and all is neighbors: g i (x i, x i ) 0 (6) wih g i : R ni Rñi R nci. Consider he following cos N v l( x, ũ) = l i (x i, u i, x i, ũ i ) (7) i=1 where l i : R ni R mi R Rñi mi R is he cos associaed o he i-h sysem and is a funcion only of is saes and he saes of is neighbor nodes. l i (x i, u i, x i, ũ i ) = l i,j (x i, u i, x j, u j ) + (i,j) A (q,r) A,(i,q) A,(i,r) A l q,r (x q, u q, x r, u r ) (8) where l i,j : R ni R mi R nj R mj R is he cos funcion involving wo adjacen nodes. Assume ha l is a posiive convex funcion and ha l i (x i e, u i e, x i e, ũ i e) = 0 and consider he infinie ime opimal conrol problem J ( x) min l( x, ũ ) (9) {ũ 0,ũ 1,...} =0 subj. o x i +1 = f i (x i, ui ), i = 1,..., N v, 0 g i,j (x i, xj ) 0, i = 1,..., N v, (10) 0, (i, j) A x i X i, u i U i, i = 1,..., N v, 0 x 0 = x

3 For all x R Nv ni, if problem (10) is feasible, hen he opimal inpu ũ 0, ũ 1,... will drive he N v sysems o heir equilibrium poins x i e while saisfying sae, inpu and inerconnecion consrains. Remar 1: Throughou he paper we assume ha a soluion o problem (10) exiss and i generaes a feasible and sable rajecory for he eam sysem. Our assumpion is no resricive. If here is no infinie ime cenralized opimal conrol problem fulfilling he consrains, hen here is no reason o loo for a decenralized receding horizon conroller wih he same properies. Remar 2: Since we assumed ha he graph is undireced, here will be redundan consrains in problem (10). Noe he form of consrains (6) is raher general and i will include he case when only parial informaion abou saes of neighboring nodes is involved. Wih he excepion of a few cases, solving an infinie horizon opimal conrol problem is compuaionally prohibiive. An infinie horizon conroller can be designed by repeaedly solving finie ime opimal conrol problems in a receding horizon fashion as described nex. A each sampling ime, saring a he curren sae, an open-loop opimal conrol problem is solved over a finie horizon. The opimal command signal is applied o he process only during he following sampling inerval. A he nex ime sep a new opimal conrol problem based on new measuremens of he sae is solved over a shifed horizon. The resulan conroller is ofen referred o as Receding Horizon Conroller (RHC). Assume a ime he curren sae x o be available. Consider he following consrained finie ime opimal conrol problem J N ( x ) N 1 min {U } =0 subj. o l( x,, ũ, ) + l N ( x N, ) (11a) x i +1, = f i (x i,, ui, ), i = 1,..., N v, 0 g i,j (x i,, xj, ) 0, i = 1,..., N v, (i, j) A, = 1,..., N 1 x i, X i, u i, U i (11b) i = 1,..., N v, = 1,..., N 1 x N, X f, x 0, = x where N is he predicion horizon, X f R Nv ni is a erminal region, l N is he cos on he erminal sae. In (11) we denoe wih U [ũ 0,,..., ũ N 1, ] R s, s N v mn he opimizaion vecor, x i, denoes he sae vecor of he i-h node prediced a ime + obained by saring from he sae x i and applying o sysem (1) he inpu sequence u i 0,,..., u i 1,. The ilded vecors will denoe he predicion vecors associaed o he eam sysem. Le U = {ũ 0,,..., ũ N 1, } be he opimal soluion of (11) a ime and J (x ) he corresponding value funcion. Then, he firs sample of U is applied o he eam sysem (3) ũ = ũ 0,. (12) The opimizaion (11) is repeaed a ime + 1, based on he new sae x +1. I is well nown ha sabiliy is no ensured by he RHC law (11) (12). Usually he erminal cos l N and he erminal consrain se X f are chosen o ensure closed-loop sabiliy. A reamen of sufficien sabiliy condiions goes beyond he scope of his wor and can be found in he surveys [9], [10]. We assume ha he reader is familiar wih he basic concep of RHC and is main issues, we refer o [9] for a comprehensive reamen of he opic. In his paper we will assume ha erminal cos l N and he erminal consrain se X f can be appropriaely chosen in order o ensure he sabiliy of he closed-loop sysem. In general, he opimal inpu u i o he i-h sysem compued by solving (11) a ime, will be a funcion of he overall sae informaion x. In [1] we have described a procedure o decompose problem (11) ino smaller subproblems whose independen compuaion can be disribued over he graph nodes. In [1] we have proposed a decenralized conrol scheme where problem (11) is decomposed ino N v finie ime opimal conrol problems, each one associaed o a differen node as deailed nex. Each node nows is curren saes, is neighbors curren saes, is erminal region, is neighbors erminal regions and models and consrains of is neighbors. Based on such informaion each node compues is opimal inpus and is neighbors opimal inpus. The inpu o he neighbors will only be used o predic heir rajecories and hen discarded while he firs componen of he opimal inpu o node i will be implemened a he i-h node, where i was compued. The i-h subproblem will be a funcion of he saes of he i-h node and he saes of is neighbors. The soluion of he i-h subproblem will yield a conrol policy for he i-h node of he form u i = f i (x i, x i ). However, he sudy in [1] does no guaranee consrain fulfillmen. In he nex secion we analyze decenralized RHC design when he nodes of he graph G have prioriies assigned o hem. We firs decompose he original G ino overlapping subgraphs wih differen hierarchy levels (Secion III). Then, we solve he problem for each subgraph independenly (Secion IV) assuming ha nodes wih high prioriies compue conrol laws for nodes wih lower prioriies. Finally, in Secion V we propose a class of hierarchical RHC conrol schemes ha guaranee sabiliy and feasibiliy. III. HIERARCHICAL DECOMPOSITION Consider he nodes of he graph G of he original cenralized problem descripion and divide i ino n g inersecing subgraphs G i = {A i, V i }, wih A = n g i=1 A i, V = n g i=1 V i and i j V i Vj. We assign o each subgraph G i a cerain prioriy, p(g i ) =. The decomposiion ino subgraphs follows a hierarchical scheme where G i has he

4 Hierarchy Hierarchy (a) (b) Fig. 1. Valid decomposiions. same or lower prioriy han he subgraphs G j wih lower indices j = 1,..., i, i.e., p(g i ) p(g j ), for j = 1,..., i. The nodes of each subgraph G i are pariioned ino wo groups: he group of predecessor nodes Vj P and he group of successor nodes Vj S, V j = {Vj P, VS j }. The decomposiion has o saisfy he following propery: for a given subgraph G i, wih i 2, he predecessor group composed of nodes Vi P is a subse of he group of successor nodes of only one hierarchically preceding subgraph for i = 2,..., n g, i.e.,!j < i Vi P Vj S. We will denoe by Pr he predecessor funcion, i.e., he funcion ha associaes o each subgraph wih index j, he index i of he subgraph which conains he predecessor nodes of G j, i.e., i = Pr(j) x Gj P x Gi S. The predecessor nodes of G 1 are hose ha are no elemens of any oher subgraph, i.e., i V1 P i / V j for j = 2,..., n g. Figures 1-2 depic examples for boh valid and invalid decomposiions. Figure 1(a) shows a decomposiion ino hree subgraphs G = 3 i=1 G i. The definiion of predecessor and successor nodes are he following: G P 1 =, G S 1 = {1, 2, 3}, G P 2 = {2}, G S 2 = {4, 5}, G P 3 = {5}, G S 3 = {6} Similarly, for he decomposiion in Fig. 1(b) we have G P 1 =, G S 1 = {1, 2}, G P 2 = {2}, G S 2 = {3}, G P 3 = {2}, G S 3 = {4} According o he noaion used in he previous secions we use he symbol ñ i,p,s o idenify he number of saes of all he nodes of he subgraph G i, ñ i,s for he number of saes of all he successor nodes of subgraph G i, ñ i,p for he number of saes of all he predecessor nodes of subgraph G i. We denoe by x i,p,s R ni,p,s he saes of he nodes of graph G i a ime, x i,p R ni,p and x i,s R ni,s are predecessor and successors saes of he nodes in G i a ime, respecively. IV. SIMPLE HIERARCHICAL SCHEME ASSUMING COMMUNICATION DELAYS Consider he sysems (1) and he inerconnecion graph G decomposed as discussed in Secion III. Consider he following finie ime opimal conrol problem for subgraph G v. (P v ) : JN v (xv,s Ũ v,p, U v,s subj. o Ũ v,s, x v,p min,u v,p ) N 1 =0 l v,s,p (x v,s,, uv,s,, xv,p,, uv,p, ) (13a) x i +1, = f i (x i,, ui, ), 0, i V v x i, X i, u i, U i, = 1,..., N 1, i V v g i,j (x i,, xj, ) 0, = 1,..., N 1, (i, j) A v x i N, X f i, i V v x i 0, = x i, i V v, Ũ v,p (13b) where {u i 0,,..., u i N 1, i V v S }, {u i 0,,..., u i N 1, i V v P } denoes he opimizaion vecors, i.e., he inpus o all he nodes of he subgraph G v grouped ino successors and predecessors. Analogously, we define Ũ v,s, {u i, i V v S }, x v,s, {x i, i V v S }, {u i, i V v P }, x v,p {x i, i V v P }, where x i, denoes he sae vecor of he i-h node prediced a ime + obained by saring from he sae x i and applying o he i-h sysem (1) he inpu sequence u i 0,,..., u i 1,. X i f Rni is a erminal consrain for he i-h node. The

5 Hierarchy Hierarchy (a) Predecessor nodes in G 3 are successors of muliple preceding subgroups G 1 and G 2. (b) G3 P is a successor of boh G 1 and G 2. G 6 has no successor nodes and G4 S, GS 5 are conneced. Fig. 2. Invalid decomposiions. cos l v,s,p is defined as l v,s,p (x v,s,, uv,s,, xv,p,, uv,p, ) = l i,j (x i, u i, x j, u j ) (i,j) A v (14) Noe ha he summaion is done only over he nodes and edges conained in he subgraph G v. We propose he following disribued sraegy. Consider a subgraph G v and he associaed problem P v (13). The opimal conrol sequence Ũ v,p for he predecessor nodes in group v are also calculaed as soluions for he successor nodes in he preceding group Pr(v). Le us modify he problem of group v by formulaing and solving a finie ime opimal conrol problem similar o (13) bu consrained o use he conrol sequence calculaed for he predecessor nodes by he preceding group. The only degree of freedom lef is o obain conrol sequences Ũ v,s for he successor nodes. This means ha nodes in Vv P will be implemening a conrol sequence Ũ v,p received from he preceding group Ũ Pr(v),S and he successor nodes will be implemening he opimal conrol soluions Ũ v,s calculaed by group v assuming a one ime sep communicaion delay. These opimal conrol soluions for he successor nodes in G v have o respec predecessor-successor consrains represened by g i,j in (13b). This sraegy can be summarized by he following algorihm. Algorihm 1: Algorihm for propagaing RHC soluions 1) For all 0 2) For all v = 1,..., n g a) Measure he saes of all he nodes in he group G v. b) Solve P 1 if v = 1, oherwise if v > 1 augmen and solve problem P v (13) wih he following 1. node 2. node 3. node Fig. 3. Solves for Solves for u [,..., N ], u [,..., N ], { Similarly, and so on... N N Transmied u [,..., N ], N u [,..., N ], { N N Transmied Propagaion of he RHC subproblem soluions. consrains Ũ v,p, Implemen firs value Implemen firs value = Ũ Pr(v),S,, = 1,..., N 1. (15) c) Transmi and implemen he soluion u i 0, on he i-h node for all i V S v. Figure 3 illusraes he propagaion of he soluion in a simple case where he nodes are conneced as a sring, following one afer anoher.

6 V. HIERARCHICAL DECOMPOSITION SCHEME WITH STABILITY AND FEASIBILITY GUARANTEES The scheme presened in he previous secion does no guaranee consrain fulfillmen. In his secion, we propose a decenralized scheme for which feasibiliy a ime zero guaranees feasibiliy a all ime insans > 0 as well as sabiliy. The main algorihm is described nex. Algorihm 2: Feasible Se Projecion Algorihm 1) Consider he eam sysem (3) and he inerconnecion graph G (4) decomposed as discussed in Secion III. 2) Consider problem P 1 in (13) wih X i f = xi e, i V 1 and compue he se X 1,P,S R n1,s,p of feasible iniial predecessor and successor saes x 1,P, x 1,S for problem P 1. 3) Compue X 1,S as he projecion of he se X 1,P,S on he successor space R n1,s, i.e., X 1,S = {x 1,S R n1,s x 1,P R n1,p, (x 1,S, x 1,P ) X 1,S,P } 4) for v = 2,..., n g, a) Define a new RHC problem ˆP v by augmening problem P v (13) wih he consrains g i,j (x i,, xj, ) 0, = 1,..., N 1, (i, j) A v, x v,p X Pr(v),S. (16) b) Compue X v,s as he projecion of he se X v,p,s on he successor space R nv,s. The algorihm presened above uses wo main conceps. Firs, i requires he compuaion of he feasible domains of he RHC problems P v in an hierarchical increasing order. Second, i ransforms he original RHC problems ino new problems ˆP v where consrains beween predecessor and successors nodes are robusly enforced for all he successor nodes belonging o he feasibiliy domains of he hierarchically preceding subproblems. The RHC conrol policy for he eam sysem is defined as follows. Algorihm 3: RHC Conrol Policy 1) For all 0 2) For all v = 1,..., n g a) Measure he saes of all he nodes in he group G v. b) Solve ˆP v if v > 1 or P 1 if v = 1. c) For all i Vv S, implemen u i 0, on he i-h node. The following heorem can be saed on he feasibiliy of he scheme presened above. Theorem 1: Consider he eam sysem (3), he equilibrium pair ( x e,ũ e ), and he consrains (2). Assume a given inerconnecion graph (4) and he inerconnecion consrains (5) using he RHC conrol policy described in Algorihm 3, where he subproblems ˆP v have been defined in Algorihm 2. If problems P 1 and ˆP v (v = 2,..., n g ) are feasible a ime = 0, hen he RHC policy described in Algorihm 3 sabilizes he eam sysem (3), in ha lim = x e lim = ũ e 1 2 x 1 x 2 x 3 Fig. 5. Simple illusraive example. while fulfilling he sae, inpu and inerconnecion consrains. Proof: The proof follows by sandard Lyapunov argumens and is omied in his version of he paper. Remar 3: The proof of Theorem 1 can be exended when erminal invarian ses and conrol Lyapunov funcions are used insead of erminal poin se consrain. Remar 4: In Algorihm 2 he sep of compuing he feasibiliy domains of problem ˆP v is no a rivial one. Algorihms are available when problems ˆP v can be cased as linear, quadraic, mixed-ineger linear and mixed-ineger quadraic programs [11]. A. Graphical Illusraion of he Mehod In his secion we describe he approach presened in he previous secion hrough a simple example. Figure 5 shows wo subgraphs G 1 and G 2, wih wo nodes in each of hem. Node 2 is he successor of group G 1 and he predecessor of group G 2. G 1 has a higher prioriy han G 2. Denoe by x 1 R, x 2 R, x 3 R he saes of node 1,2 and 3 respecively. The RHC problem associaed wih G 1 and G 2 will be denoed by P 1 and P 2, respecively. The sae feedbac soluion P 1 is compued firs and i is a funcion of x 1 and x 2. We denoe by u 1 and u 2 he sae feedbac soluion of problem P 1 for node 1 and 2, respecively: u 1 = f 1 (x 1, x2 ), u2 = f 2 (x 1, x2 ). X 1,P,S R 2 is he domain of he funcions f 1 and f 2. When solving problem P 2, he nowledge of X 1,P,S allow us o guaranee ha he inpus compued for x 3 will be feasible for he closedloop behavior of node 2. We follow he seps described in Algorihm 2. We compue he projecion of X 1,P,S on he successor space, and obain a se in 1 dimension denoed by X 1,S. Consider now problem P 2 and he inerconnecion consrains g 2,3 (x 2, x3 ) 0, 0 (17) We saisfy consrain (17) for all he saes x 2 which are feasible a he higher level. Tha is, we consruc a new problem ˆP 2 where consrain (17) is subsiued wih g 2,3 (x 2, x3 ) 0, x2 X 1,S, 0 (18) Problem P 2 is sill funcion of he sae x 2 because i eners he cos funcion.

7 x 1 x 1 x 1 1,S 1,S 1,S 2,S x 2 2,S = Ø x 2 2,S x 2 Fig. 4. Simple examples illusraing he Feasible Se Projecion Algorihm. The shaded areas denoe g 2,3 (x 2, x 3 ) 0. Clearly he consrain (18) can be infeasible and such approach migh be conservaive. However he nowledge of he ses X 1,S allows us o reduce he degree of conservaiveness wih respec o oher approaches which consider all possible behavior of he saes x 2. Also, noe ha only he nowledge of he feasible ses is required and no of he sae-feedbac laws f 1 and f 2. Consider he example above and he hree cases depiced in Figure 4. We assume o have solved P 1 and o have compued X 1,S (he bold lines in he x 2 space). The shaded areas depic hree possible feasible ses described by he inerconnecion consrains g 2,3 (x 2, x 3 ) 0. A quic glance a Figure 4 ells ha he conservaiveness of he proposed mehod depends on he shape of he feasible region described hrough he inerconnecion consrains. In he firs case, problem ˆP 2 is always feasible, in he second case ˆP 2 is infeasible and in he hird case ˆP 2 is feasible over wo disconneced ses. A possible way of reducing conservaiveness is o inroduce communicaion of he admissible feasible regions beween he hierarchical levels, he wor in [12] is a preliminary sudy in his direcion. We are currenly looing a differen proocols for exchanging bounds on he feasible domains. [3] V. D. Blondel and J. N. Tsisilis, A survey of compuaional complexiy resuls in sysems and conrol, Auomaica, vol. 36, no. 9, pp , [4] M. Roowiz and S. Lall, Decenralized conrol informaion srucures preserved under feedbac, in Proc. 41h IEEE Conf. on Decision and Conrol, [5] R. D Andrea and G. Dullerud, Disribued conrol of spaially inerconneced sysems, IEEE Trans. Auomaic Conrol, o appear. [6] K. Gobayra and C. Cassandras, A hierarchical decomposiion mehod for opimal conrol of hybrid sysems, in Proc. 38h IEEE Conf. on Decision and Conrol, Phoenix, AZ, December 1999, pp [7] C. Cassandras, D. Pepyne, and Y.Wardi, Opimal conrol of a class of hybrid sysems, IEEE Trans. Auomaic Conrol, vol. 46, no. 3, pp , [8] D. Sipanovic, G. Inalhan, R. Teo, and C. Tomlin, Decenralized overlapping conrol of a formaion of unmanned aerial vehicles, in Proc. 41h IEEE Conf. on Decision and Conrol, [9] D. Mayne, J. Rawlings, C. Rao, and P. Scoaer, Consrained model predicive conrol: Sabiliy and opimaliy, Auomaica, vol. 36, no. 6, pp , June [10] D. Mayne, Conrol of consrained dynamic sysems, European Jornal of Conrol, vol. 7, pp , [11] F. Borrelli, Consrained Opimal Conrol of Linear and Hybrid Sysems, ser. Lecure Noes in Conrol and Informaion Sciences. Springer, 2003, vol [12] A. Richards and J. P. How, A decenralized algorihm for robus consrained model predicive conrol, in Proc. American Conr. Conf., VI. EXAMPLES In he final version of he paper we will inroduce wo examples where we compare he simple scheme wihou guaranees presened in Secion IV wih he scheme presened in Secion V for he formaion flying scenario [1]. The echnique presened in his paper can be very effecive for such applicaions. In fac, even if he feasible domains X v,p,s are non-convex in general (because of he nonconvexiy of he collision avoidance consrains), heir projecions X v,s on he successors are convex. Thus, robus consrain fulfillmen can be achieved wihou addiional compuaional effor. REFERENCES [1] T. Keviczy, F. Borrelli, and G. J. Balas, A sudy on decenralized receding horizon conrol for decoupled sysems, in Proc. American Conr. Conf., [2] S. Wang and E. J. Davison, On he sabilizaion of decenralized conrol sysems, IEEE Trans. Auomaic Conrol, vol. 18, no. 5, pp , 1973.