Known unknowns, unknown unknowns and information flow: new concepts in decentralized control

Known unknowns, unknown unknowns and info rmation flow: new concepts and challenges in d ecentralized control M.-A. Belabbas Abstract — W e introduce and analyze a model for decentral- ized control. The model is broad enough to include problems such as forma tion control, decentralization of the p ower grid and flocking. The objectiv e of thi s paper is twof old. F irst, we show how the issue of decentralization goes beyond having agents k now onl y part of the state of th e system. In fact, we argue th at a complete th eory of decentralization shoul d take into account the fact that agents can be made awar e of only part of th e global objectiv e of the ensemble. A second contribution of this paper is the introduction of a rigorous defin ition of informa tion flow fo r a decentralized system: we show how to attach to a general n onlinear decentralized system a un ique information fl o w graph that is an in variant of the system. In order to ad dress some fi ner issu es in decentralized system, such as the existence of so-called ”inf ormation loops”, we f urther refine the inform ation flow g raph to a simplicial complex—more precisely , a Whitney complex. W e illustrate the main results on a variety of examples. I . I N T R O D U C T I O N Inform ally sp eaking, a dec entralized contro l system is a system whose d ifferent parts—let us call the different parts agents—are not told what to do by a unique, centralized controller, but de cide what to d o based on th e possibly incomplete in formation that is at their disposal. The importanc e o f decentralization in control has been recogn ized f or ma ny decade s [1], but o nly in more r ecent time h as the issue been the sub ject of sustained investigation, see [2], [3], [4], [5], [6], [7] and refe rences therein. T his renewed in terest is fu elled, on the one hand, by the potential a complete theory of decentralize d co ntrol has in explaining natural behavior: flocks of b irds, ant co lonies, etc. and more broadly by its application s in decision theo ry [ 8] and cogni- tion. On the o ther ha nd, a theor y o f d ecentralizatio n is also a necessity for engineering d esign: from s mart grids to v ehicles managem ent on the high way [9], [10], r ecent developments in rob otics and commu nication h av e made it p ossible to en vision very la rge gr oups of auto nomo us vehicles or ag ents collaboratin g to ach iev e a glo bal objec tiv e. In this context, decentralizatio n is thus necessary for reasons rang ing from robustness— failure a t some level ( agent, contro ller , etc.) in a centralized system is likely to affect the en tire system, whereas failure in a d ecentralized system is mo re easily handled —to, at a mo re fundam ental le vel, f easibility . Indeed , a cen tralized contro ller for, say , vehicles on th e high way is not e asily implemented . The objec ti ve of this p aper is twofold. First, we will show that the idea o f inco mplete information in a de centralized M.-A. Belabb as is with the School of Engineering and Applied Scien ces, Harva rd Univ ersity , Cambridge , MA 02138 belabbas@seas .harvard.edu setting should be explored beyo nd the usual pa rtial knowl- edge o f the state of the system. While most of the extant work in decentra lized control implicitly assume that all the agents know the objective of the ensemble and are thu s solely constrained by their limited ob servations, we develop h ere a model which includes r estrictions on what agents k now ab out the global objective of the ensemble. W e illustrate this id ea on a form ation contro l prob lem in [11] by showing that local stabilization aroun d a g iv en c onfigur ation is possible o nly if agents k now more than the ir selfish o bjective. Second, we will make rig orous the notio n of informa- tion flow in a decentralized system . Th e naive notion of informa tion flow that is often used in the linear theory of decentralized system—i.e. splitting variables into g roups and coding the d ependen ce be tween group s by a gr aph, see Section IV—is inher ently depende nt on the ch oice of coordin ates used. This aspe ct puts it at o dds with the idea that what an agent knows abo ut th e system should not depend o n th e way one c hooses to describ e the system. Even more, the naive information flow does not ackn owledge the possibility that Lie bra ckets may be needed to make the system controllab le. While the se issues can b e sidestepped in the line ar case to obtain results that are nevertheless meaningf ul, they become a g enuine limitation whe n one tries to understan d nonlin ear decentralized sy stems or systems with con straints. I ndeed, in the f ormer situation, there of- ten do es not exist p referred coor dinates or one may need se veral coord inate c harts to describe the system. In the latter situation, choosing coord inates that are compa tible with the constraints, e.g. a c onservation of en ergy con straint, changes the n aiv e inf ormation flow since it often requ ires a m ixing of th e coordin ates used to describ e the age nts ind ividually . W e introd uce in th is paper a definition of info rmation flow grap h that is in variant u nder changes of coordinates and allows to define rigoro usly decentralizatio n in a contr ol system. The main id ea b ehind this d efinition is th at the observation function s on the system provid e a na tural set of vertice s for the inform ation flow graph. The paper is organized as fo llows. W e first intr oduce a general no nlinear mo del for d ecentralized c ontrol systems. W e then define th e globa l o bjective of a system and th e local or selfish ob jectives of the agents. The fact that th e agents only kn ow pa rt of the glo bal objective is en forced throug h the use of n on-invertible fu nctions on the pa ram- eters d escribing the g lobal o bjective; this a pproac h can be understoo d, inf ormally speakin g, as a decentralization of the objective or a decentraliza tion of the d esign o f the contro l law . In the f ollowing section, we introd uce a partial order on the local objectiv es and observation functions. Th is partial order allows u s to qu antify the idea tha t some local objec- ti ves (resp. o bservations) are more revealing of the global objective ( resp. state of the ensemble) than o thers. In Section IV, we intro duce a co ordin ate free de finition of informa tion flow in a d ecentralized system. Motivated by the existence of ”inform ation loops” in d ecentralized system, we further refine the in formatio n flow graph into an inform ation flow comp lex tha t re veals finer issues in d ecentralization . Example 1 (Power grid) . In r ec ent years, the development of me thods to insur e the stability of the power grid have come at the for efr ont of r esea r ch in contr ol theory . I n this context, one can view the power grid as a very lar ge scale system whose glo bal ob jective is to r emain stable ar oun d a desir ed operating point. Such systems currently operate in a centralized manner un der th e su pervision o f Supervisory Contr ol and Data Acquisition system (SCAD A ). This centralized framework, however , is starting to show its limitations due to the increasingly complex compo nents that ar e pa rt of the grid (e.g. gr een energy suppliers). The development of decentralized methods to insure the stability and g ood operation of the grid have thus b ecome a p riority in po wer systems engineering. In this context, the global o bjective is a function of all the compone nts of the grid, but it is clearly no t feasible to let all agents in th e grid know a bout its co mplete ar chitecture . Example 2 (Formation contro l) . Let x i ∈ R 2 r epresent the positions of autono mous agents in the plane a nd d i ∈ (0 , ∞ ) be real po sitive con stants. W e co nsider the formation contr ol pr oblem whose dynam- ics a r e given by ˙ x 1 = e 1 ( x 2 − x 1 ) + e 5 ( x 4 − x 1 ) ˙ x 2 = e 2 ( x 3 − x 2 ) ˙ x 3 = e 3 ( x 1 − x 3 ) ˙ x 4 = e 4 ( x 3 − x 4 ) wher e we denote by e i the err or in edge leng th: e 1 = k x 2 − x 1 k 2 − d 1 , e 2 = k x 3 − x 2 k 2 − d 2 , . . . , e 4 = k x 3 − x 4 k 2 − d 4 , e 5 = k x 4 − x 1 k 2 − d 5 The information flow o f the system is r epresented in F igure 1a. F ormation contr ol pr o blems ar e defined up to a rigid transformation of the p lane [12]. F or this r easo n, one often describes the d ynamics in terms of the inter-agent distances [13]            z 1 = x 2 − x 1 z 2 = x 3 − x 2 z 3 = x 1 − x 3 z 4 = x 3 − x 4 z 5 = x 4 − x 1 , (1) instead of the ab solute position s of the agents. In the z variables, the dyna mics o f the system a r e g iven x 1 x 2 x 3 x 4 (a) Nai ve information flow for the x va riable s z 1 z 2 z 3 z 4 z 5 (b) Nai ve in formation flo w for the z v ariabl es Fig. 1: The naiv e information flow o f a system is a d irected graph wh ose vertices v i correspo nd to grou ps of variables describing the system. There is a dir ected edg e fro m v i to v j if the dynam ics of variables in the group of v i depend on the variables in the group of v j . This graph depends o n the coordin ates chosen to d escribe the system. by ˙ z 1 = e 2 z 2 − e 1 z 1 − e 5 z 5 ˙ z 2 = e 3 z 3 − e 2 z 2 ˙ z 3 = e 1 z 1 + e 5 z 5 − e 3 z 3 ˙ z 4 = e 3 z 3 − e 4 z 4 ˙ z 5 = e 4 z 4 − e 1 z 1 − e 5 z 5 The corr espondin g naive information flo w ha s 5 vertices and is r ep r esen ted in Figur e 1 b. W e d escribe in Section IV a way to ob tain the informa tion flow depicted in F igure 1 a fr om the description of th e system given in terms of the z variables. I I . A M O D E L F O R D E C E N T R A L I Z E D C O N T RO L W e pr esent in this section a gener al mo del for n onlinear decentralized control systems. T he model is a natural exten- sion of the n otion of decentralized system that is en counter ed in the literatu re o n lin ear sy stems. In addition to being applicable to nonlinear problems, such as formatio n co ntrol, our app roach distinguishes itself fro m most o f the work on linear decentralized co ntrol in at lea st two ma jor a spects [2], [3], [5]: - it in troduce s the notio n of p arametrized o bjective of a decentralized system. - it allows for loop s of information in the system, unlike approa ches b ased on qu adratic in variance or par tial orders [2 ], [3]. W e revisit this p oint in Section IV. A. General mo del Let M be a smooth m anifold and the state x ∈ M . W e consider n onlinear control systems of the type ˙ x = f ( x, u ( x )) = n X i =1 u i ( δ i ( µ ); h i ( x )) g i ( x ) (2) where δ, h are smooth functions, th e g i ’ s are smooth vector fields an d µ is a para meter that describ es the objective of the system. W e let U be the space of a dmissible co ntrols u i . W e elaborate o n th e various parts of the model in this section. A c ommon situation is f or th e manifold M to be the produ ct of the manifolds describing the state-spac es o f each agent: M = n O i =1 M i where M i is the state-space of agent i . W e can th us write that the tangent sp ace o f M is the direct sum T M = ⊕ i T M i . In this case, we also have that the p rojection of g i ( x ) on to T M j is zer o if i 6 = j : π j g i ( x ) = 0 if i 6 = j. This product structure is often lost due to either interaction s between agents which impose constraints on the state x ∈ M , or the existence of a symmetry grou p acting o n M , in which case on e has to co nsider eq uiv a lence classes o f states x ∈ M (this is the case in, e.g., f ormation co ntrol). Hen ce we do not assume h ere any special structure f or M . W e differentiate between two typ e of objectives: 1) the ob jectiv e that each ag ent or plant tries to satisfy: it is r eferred to a s lo cal objective or selfish objective. 2) the ob jective the agen ts try to ac hieve by coop erating: it is r eferred to a s g lobal ob jectiv e or comm on objective. The functio ns δ i in Equation (2) allow us to contro l how much an agent knows about the common objectiv e of the ensemble; in some sen se, these f unction s intro duce a partial observation on th e o bjective o f the ensemble, a kin to the partial o bservation that the agen ts have on the state of the ensemble. Th ey are described in more detail b elow; w e start with the de finition of loc al ob servations. B. Local observations The m ain character istic of a decentralized control system is that the agen ts are only ab le to observe p art of the state of th e system. W e introdu ce the functions h i ( x ) : M → R k i , k i a positiv e integer to describe the observation of agent i on the current state of the system. W e denote by h i ( M ) the image of M un der the map h i . C. Local and g lobal objectives W e define in this section the local a nd glo bal ob jective of a decentralized system . W e con sider the case of ob jectives that depe nd on a par ameter . This level of generality is often necessary to ac curately model dece ntralized systems whose dy namics are rich eno ugh to accomm odate parameter- varying objectives, such as flocks o f au tonom ous agents or power distribution systems. 1) Globa l ob jective: Let P be a smoo th m anifold, we let µ ∈ P param etrize th e glo bal ob jectiv e of th e co ntrol system as f ollows: Definition 1 (Globa l o bjective) . Given a decentralized con - tr ol system ˙ x = f ( x, u ( x )) of the type of Equatio n (2) , the global objective fu nction is a differ entiab le function F ( µ ; x, u ) : P × M × U → R d with the convention that th e o bjective is a chieved if the system is a t x ∗ ∈ M with F ( µ ; x ∗ , u ) = 0 for eq uality obje ctiv es or F ( µ ; x ∗ , u ) ≥ 0 for ineq uality ob jectiv es , where the inequ ality is taken entry- wise. The objective functio n can in gen eral d epend on u ; this depend ence is n ecessary if on es con siders stabilizatio n ob- jectiv es. When a g lobal ob jectiv e is n ot p arametric or does not dep end o n u explicitly , we omit the d epende nce from the notation. W e give a few examp les: Example 3 ( Rendez-vous) . Consider a multi-agent system with two agents whose positions are given by x 1 ∈ R m and x 2 ∈ R m . Th e g lobal objective is to have the agents meet. This o bjective does not dep end on a parameter . W e can encode it by F ( x 1 , x 2 ) = −k x 1 − x 2 k 2 . If we want the agents to r ea ch a p osition such that they ar e at a given d istance d fr om each othe r , we let P = [0 , ∞ ) and we u se F ( d ; x 1 , x 2 ) = − ( k x 1 − x 2 k 2 − d 2 ) 2 . Example 4 (Stabilization) . Con sider the simple nonparamet- ric r endez-vou s pr oblem describe d in the pr evious example with the a ddition that the agents ar e r eq uir ed to stabilize at the r end ez-vous co nfiguration. W e d enote b y ∂ f ∂ x | x ∗ the J a cobian o f the system at x ∗ . W e de note by λ i ( A ) the eigen va lue o f A with i th lar gest real part. W e can repr esent this global ob jective b y using the vector-valued fun ction F ( d ; x, u ) : R m × R m × U → R m +1 : x →      − ( k x 1 − x 2 k 2 − d 2 ) 2 − Re ( λ 1 ( ∂ f ∂ x )) . . . − Re ( λ n ( ∂ f ∂ x ))      (3) 2) Local Objectives: W e now fo cus o n describing the system at the le vel o f the agen ts. W e d efine a local o bjective as being, roughly speaking, a restriction of a global objective. Definition 2 (Local objec ti ve) . Given a decentralized contr ol system with globa l objective parametrized by P , we let P i be a smoo th ma nifold a nd δ i : P → P i be smoo th functio ns. F or µ ∈ P , the local o bjective of agent i is g iven b y a smoo th fu nction F i ( δ i ( µ ); h i ( x ) , u i ) : P i × h i ( M ) × U i → R d with the same convention as in Defin ition 1 r egar ding equal- ity and in equality o bjectives. When δ i is not an invertible function, an agent knows about part o f the g lobal objective . W e give some examples of relations between global and local objectiv es in the section below . The decen tralized co ntrol p roblem is well-posed if satis- fying the loc al objectives is sufficient to satisfy the globa l objective: f i ( δ i ( µ ) , h i ( x ) , u i ( x )) ≥ 0 for all i = ⇒ F ( µ, x, f ( x )) ≥ 0 and similarly fo r th e equality ob jectiv e. A wide array of questions in decentralized control can the n be r educed to o ne on the following three major questions: 1) How little information can we let th e agents know about the g lobal objective and still have the ensemble achieve it? In other words, how informative do we need the δ i to be in or der to achieve a gi ven global objective? 2) Gi ven a glo bal objective, how little observation on the system d o th e age nts nee d in o rder to achieve the globa l objective? In other words, how inf ormative do th e h i need to be in ord er to achieve a g lobal ob jectiv e? 3) Gi ven a decentr alized system with fixed o bservation function s h i ( x ) an d con trol vector field s g i ( x ) , wh at global objectives are achiev able? The fir st two qu estions are not indepe ndent. Indeed, if the observation fun ctions h i ( x ) d o not provide much informatio n about th e state o f th e en semble, increasing the knowledge an agent has abo ut th e glob al ob jectiv e is likely to be fru itless (a ty pical examp le is fo rmation con trol). W e introdu ce below a par tial order on observations and o bjective that allow us to attach a math ematically p recise me aning to these questions. Example 5 (Formations) . Conside r a formation contr o l pr oblem where n agents in the pla ne, with position s x i ∈ R 2 , ar e r equ ir ed to stabilize a t the configuration described in F igure 2. This co nfigu ration is d efined up to a tr anslation and r otation of th e pla ne. W e h ave shown in [12] that the space of such confi gurations was C P( n − 2 ) × (0 , ∞ ) . Hence, the p arameter space is P = C P ( n − 2) × (0 , ∞ ) . A po int in P can also be r epr esen ted, up to mirr or symmetry , by the inter-ag ent distances [ d 1 , . . . , d N ] , wher e N = 1 2 n ( n − 1) . The d i ’ s ar e of co urse r ed unda nt in this r epr esen tation, since simple trigonometric rules r elate th e pairwise distances. The pr oblem of finding a n on-redundant repr esentation based on fewer pairwise distances is r elated to globa l rigidity [14 ] It may be impractical, or in some situation undesirable, to let every agent know a bout the complete vector µ . The δ i intr odu ced here allow the study o f systems where the amoun t of knowledge an agent gets about µ is con tr olled. x 12 x 14 x 22 x 24 x 32 x 34 x 42 x 44 x 52 x 54 x 21 x 25 x 31 x 35 x 41 x 45 x 51 x 55 x 13 x 23 x 33 x 43 x 53 x 63 (a) x 12 x 14 x 22 x 24 x 32 x 34 x 42 x 44 x 52 x 54 x 21 x 25 x 31 x 35 x 41 x 45 x 51 x 55 x 13 x 23 x 33 x 43 x 53 x 63 (b) Fig. 2 : Con sider the f ormation co ntrol p roblem where ag ents with position s x ij ∈ R 2 are req uired to stabilize at the configur ation depicted above. W e let µ b e the vector of all pairwise distances b etween agents at the desired config ura- tion. W e represent the function δ i ( µ ) by a g raph with a v ertex per agent, an d an edge betwee n th e vertices x ij and x kl if agents x ij and x kl know the distance k x ij − x kl k at th e desired configuratio n. In or der to have the agents cooperate to stabilize at th is con figuration , we can let ea ch agent k now about the complete vector µ or on ly parts o f it. For example, letting each ag ent know about the distance to its near est neighbo rs, as sh own above in ( a ) , allows them to reconstru ct the desired configur ation. In ( b ) , the agents are gi ven less informa tion a bout the g lobal ob jective than in ( a ) , and one can see th at respecting the p airwise d istances depicted is n ot sufficient to re construct th e desired config uration. Hence the δ i in ( b ) a re n ot inf ormative eno ugh. I I I . P A RT I A L O R D E R S A N D D E C E N T R A L I Z AT I O N In the design of a decentralized co ntrol system, a glob al objective can be ach iev ed b y the use o f different ob servation function s and dif ferent lo cal objecti ves. W e put a partial order on the local ob jectiv es ( resp. observations) to form alize the notion th at different local ob jectiv es (resp. ob servations) can be m ore o r less revealing o f the globa l objective or system (resp. state o f the en semble). W e start with the de finition of partial ord er: Definition 3 (Partial order) . A partial or der  over a set F is a bin ary r e lation on the elements o f F which satisfies, fo r f 1 , f 2 , f 3 ∈ F 1) r eflexivity: f 1  f 1 . 2) antisymmetry: if f 1  f 2 and f 2  f 1 then f 1 = f 2 3) transitivity: f 1  f 2 and f 2  f 3 then f 1  f 3 The elem ents f 1 and f 2 in F are called compar able if either f 1  f 2 or f 2  f 1 hold. The set F is called a poset for p artially or dered set . An elem ent f ∈ F is a greatest (resp. sma llest) elem ent if f  f i (resp. f i  f ) f or all f i ∈ F . If a gre atest (resp. smallest) elemen t exists, it is unique. A maximal (re sp. min imal) elemen t f is such that th ere is n o f i ∈ F such that f i  f (resp. f  f i ). Remark 1. W e mention h er e th at partial or ders h ave, quite inter esting ly , b een app lied to decentralized contr ol in pr evi- ous work [3 ]. However , the appr oach and o bjective a r e quite differ ent fr om ours. In the work [3], the authors give an anal- ysis of linea r systems who se information flo w graph—we will define it in Sectio n I V—is given b y a Hasse diagram [1 5]. These are a type o f d irected acyclic graphs. They sh ow in particular , r elyin g o n [2], that ther e is a parametrization of such systems in which stab ilization q uestions can be r ed uced to conve x pr oblems. A. P artial order on δ i Recall that the function s δ i allow us to define decentralized systems whe re the ag ents k now only part of th e globa l objective of the en semble. W e further refine this n otion of incomplete knowledge of the global objective by establishing a partial order of the f unction s δ i . Let µ ∈ P , where P is a smo oth compac t man ifold. From Whitney’ s embed ding theorem [16], we k now th at for n large enoug h, we can smoothly embe d P in R n . Hence, witho ut loss of g enerality , we can assume tha t all the δ i map into R n . W e denote by N δ ( µ ) the isolevel set N δ ( µ ) = { x ∈ P s.t. δ ( x ) = δ ( µ ) } . Many function s δ i describe a similar restriction of the objective. For example, if δ i maps to R , tran slating the function by a constant c ∈ R to δ i ( µ ) + c d oes not change, for all p ractical p urposes, wha t ag ent i knows abo ut the global objective. I ndeed, if the ob jectiv e is realize d with the control u i for δ i , it is realized with the co ntrol ˜ u i ( δ i ; x ) = u i ( δ i − c ; x ) for δ i + c . W e generalize this idea in the following definition: Definition 4 (Eq uiv a lence of local objectives) . The fu nctions δ 1 : P → R n and δ 2 : P → R n ar e equiva lent at µ ∈ P , written δ 1 ≈ µ δ 2 , if N δ 1 ( µ ) = N δ 2 ( µ ) . They ar e equ ivalent if the above is true for all µ ∈ P . Hence two functions are equiv alent if their isole vel sets are the same. I n particu lar , all one-to-o ne in vertible fun ctions are equiv alent. This definition indeed corresp onds to the intuitive notion of eq uiv a lent k nowledge of th e glob al objective: as explained ab ove, if the isole vel s ets of δ 1 and δ 2 are the same, one can realize the same decentr alized system by using an approp riately modified control u . Example 6. Assume that we have a 2 agent decentralized system in R m wher e th e objective is parametrized by a point in P = [0 , 1] × [0 , 1] . Let µ = ( µ 1 , µ 2 ) ∈ P and δ 1 ( µ ) = µ 1 , δ 2 ( µ ) = µ 2 . The uncertainty the first agent has ab out the global o bjective is its unc ertainty abo ut µ 2 . In particular , if agent 1 ha s acce ss to ˜ δ 1 ( µ ) = µ 2 1 , it h as the same kn owledge about the globa l objective th an with δ 1 ( µ ) . Accor ding to Definition 6, δ 1 ≈ ˜ δ 1 . Observe that this would not be true if P = [ − 1 , 1 ] × [ − 1 , 1] . W e can now define a partial order o n the local objectives . Definition 5 (Partial ord er on δ i .) . Let F be the set of continuo us functio ns on P with the equiva lence r elation of Definition 4. W e say that δ 1  µ δ 2 if N δ 1 ( µ ) ⊆ N δ 2 ( µ ) . If the above r ela tion is v alid for all µ ∈ P , we simply write δ 1  δ 2 . This definition expresses the notion that δ 1 contains more informa tion than δ 2 —written as δ 1  δ 2 — if the uncertain ty arising from knowing δ 1 ( µ ) is smaller than the on e ar ising from k nowing δ 2 ( µ ) , where u ncertainty is qu antified by the isolev el sets of δ . The partial ord er  has a smallest element: the con stant function . This corresp onds to the in tuitiv e idea th at if δ i is constant, th e agent k nows nothing abou t the glo bal objective. The function s with th e h ighest level of info rmation are the in vertible fun ctions of µ ; these function s ar e the maximal elements. Example 7 . Assume th at P = S 3 = { x ∈ R 4 s.t. x 2 1 + x 2 2 + x 2 3 + x 2 4 = 1 } . W e let δ 1 ( x ) = x 1 and δ 2 ( x ) = ( x 1 , x 2 ) .W e have tha t δ 1  δ 2 . W e also ha ve δ 2 ≈ ( x 2 , x 1 ) ≈ ( x 1 + a, x 2 + b ) , for a, b ∈ R B. P artial order on h i W e similarly define a partial o rder on the ob servations h i . W e d enote by N h i ( x ) the subset o f M such that h i ( y ) = h i ( x ) : N h ( x ) = { y ∈ M s.t. h ( y ) = h ( x ) } In words, it is the set of co nfiguratio ns that are undistin- guishable to the obser vation fu nction h i . Similarly to Defin ition 4, we say that two observation function s h 1 and h 2 are eq uiv a lent if their isolevel sets on M —or th e configu rations th at are und istinguishab le for h 1 and h 2 —are th e same: h 1 ≈ h 2 ⇔ N h 1 ( x ) = N h 2 ( x ) , ∀ x ∈ M . Furthermo re, we can use a similar partial or dering on the observation function s to the one o f Defin ition 5: h 1  x h 2 if N h 1 ( x ) ⊆ N h 2 ( x ) . If th e above relation is valid for all x ∈ M , we simply write h 1  h 2 . C. P artial or der on th e f i W e furth er defin e a p artial order on the f i . In the case of equality ob jectiv es, the defin tions are similar to the o nes we have introduces above. W e thus treat h ere the case of ineq uality objectives. W e denote by N + f i ( δ i ; h i ,u i ) ( µ ) the subset of M i such th at f i ( δ i ( µ ); h i ( x ) , u i ( x )) ≥ 0 : N + f i ( δ i ; h i ,u i ) ( µ ) = { x ∈ M s.t. f i ( δ i ( µ ); h i ( x ) , u i ( x )) ≥ 0 } In words, it is the set of configuratio ns that satisfy the loca l objective f i . W e thus introduce the equ iv alenc e relation Definition 6. Th e fun ctions f 1 and f 2 ar e equ ivalent at µ ∈ P if f 1 ≈ + µ f 2 ⇔ N + f 1 ( δ i ,h i ,u i ) ( µ ) = N + f 2 ( δ i ,h i ,u i ) ( µ ) . They ar e equ ivalent if the above is true for all µ ∈ P . W e say tha t f i  µ f j if N + f i ( δ i ,h i ,u i ) ( µ ) ⊆ N + f j ( δ i ,h i ,u i ) ( µ ) . Hence, f 1  f 2 if the local objective f 1 is more stringent than the local ob jectiv e f 2 . D. Minimally in formed decentralized contr ol and saturation W e can now define Definition 7 (Min imally inform ed dec entralized system) . Given a glob al objective F , we say th at a decentralized contr ol system of the type of Eq uation (2) is minimally informed if the r e is no set of fun ctions ˜ δ i , ˜ h i , ˜ f i with δ i  ˜ δ i , h i  ˜ h i , f i  ˜ f i and such that the d ecentralized system ˙ x = X i u i ( ˜ δ i ( µ ) , ˜ h i ( x )) g i ( x ) with local ob jectives ˜ f i satisfies the glo bal obje ctive F . When lookin g f or a minim ally in formed system, an im- portant notion that arises is th e one of saturation. Since the agents only hav e access to partial observations o n the system, knowing an incr easingly larger part of µ may cease to be helpful. W e say that δ i saturates h i if fo r all ˜ δ i  δ i , ther e are no ˜ f i ( ˜ δ i ( µ ); h i ( x )) with ˜ f i  f i . Reciprocally , we hav e that h i saturates δ i if for all ˜ h i  h i , there are no ˜ f i ( δ i ( µ ); ˜ h i ( x )) with ˜ f i  f i . Example 8. Consider agent 1 in th e two-cycles formation of F igur e 3. W e let µ denote a tar get formation as explained in Examp le 5. W e let δ 1 ( µ ) = [ d 1 , d 5 ] . In the case o f range only measur eme nts h 1 ( x ) = [ k x 2 − x 1 k , k x 4 − x 1 k ] , and the loc al o bjective is given by f 1 ( δ 1 ( µ ); h 1 ( x )) =  k x 2 − x 1 k − d 1 k x 4 − x 1 k − d 5  . It is easy to see that h 1 is saturated by δ 1 . Similarly , fo r agent 2 with h 2 ( x ) = [ k x 2 − x 1 k ] , x 1 x 2 x 3 x 4 d 1 d 2 d 3 d 4 d 5 (a) x 1 x 2 x 3 x 4 d 1 d 2 d 3 d 4 d 5 (b) Fig. 3: In the two-cycles form ation dep icted above, the agents are required to stabilize at the inter-agent d istances d 1 , . . . , d 5 . Up to mirror symmetry , there are two config u- rations in the plane that satisfy these interagent distances. If agent 1 can measure the r elative positions o f ag ents 2 and 4, it can make use of the an gle betwe en the vectors ( x 2 − x 1 ) and ( x 4 − x 1 ) at the d esired configu rations ( a ) and ( b ) . This observation fu nction is thu s n ot satu rated by d 1 , d 5 . If agent 1 can only measure its distance to ag ent 1 and agent 2, the kn owledge of the angle is n ot helpful. This observation function is saturated by d 1 , d 5 . δ 2 ( µ ) = [ d 2 ] and f 2 ( δ 2 ( µ ); h 2 ( x )) = k x 2 − x 1 k − d 1 , we see that δ 2 ( µ ) = d 2 saturates h 2 . If we let h 1 ( x ) =  k x 2 − x 1 k , k x 4 − x 1 k , ( x 2 − x 1 ) T ( x 4 − x 1 )  , i.e. the first agent observes its relative distances to agents 1 a nd 4 as well as their r elative positions, then add itional knowledge o f µ is helpful. Indeed , we can pr ove in this case that h 1 is satu rated by δ 1 ( µ ) = µ . Intuitively , kn owing a ll these distan ces a llows agent 1 to establish what the possible angles between x 2 − x 1 and x 4 − x 1 ar e when the glo bal objective is reached. See [11 ] fo r a ddition al details. I V . I N F O R M AT I O N FL O W G R A P H The ab stract idea of infor mation flow , because it allows to grasp the conn ectivity of the agents a nd identify poten tial difficulties in the d istribution of info rmation in th e ensem ble, appears quite f requen tly in work on decentr alized control. W e n ow define what we inf ormally call the n aive info rma- tion fl ow of a system ; which is ofte ntimes implicitly defined in work o n decentralized contr ol. Let e j be the vecto r with zero entries except for the j th entry , which is one. Th e e j ’ s form the canon ical b asis of R n . In the case of multi- agent systems in R n , h i ( x ) will often be the pro jection of x ∈ R n onto a th e subspace spanne d b y some vectors e j , j ∈ J i where J i is a set of indices. For this reason, the observation functions h i are enco ded as a grap h with vertices x i and an edge from x i to x l if l ∈ J i . W e call this the naive inf ormation flow of the sy stem, since it is co ordinate depend ent as we illustrate in the examples b elow . Example 9 (Four ag ents) . Consider the system with M = R 4 m , x i ∈ R m and who se dynamics is given by x 1 x 2 x 3 x 4 (a) z 1 z 2 z 3 z 4 (b) Fig. 4 : The two gr aphs above represent the n aive information flow of the same system expr essed in two d ifferent coo rdinate systems. ˙ x 1 = u 1 ( x 1 , x 2 ) ˙ x 2 = u 2 ( x 2 , x 3 ) ˙ x 3 = u 3 ( x 3 , x 4 ) ˙ x 4 = u 4 ( x 4 , x 1 ) Hence, J 1 = { 1 , 2 } , J 2 = { 2 , 3 } , J 3 = { 3 , 4 } , J 4 = { 1 , 4 } . One can associate the graph of F igur e 4 a to this system, which shows a no n-trivial loo p in the in formation fl ow . Now consider the linea r chan ge of va riables: z 1 = x 1 + x 2 + x 3 + x 4 z 2 = x 2 + x 3 + x 3 z 3 = x 3 + x 4 z 4 = x 4 W e the n h ave x 1 = z 1 − z 2 x 2 = z 2 − z 3 x 3 = z 3 − z 4 x 4 = z 4 and for appr op riately defined ˜ u i , ˙ z 1 = ˜ u 1 ( z 1 , z 2 , z 3 ) ˙ z 2 = ˜ u 2 ( z 2 , z 3 , z 4 ) ˙ z 3 = ˜ u 3 ( z 3 , z 4 ) ˙ z 4 = ˜ u 4 ( z 1 , z 2 , z 4 ) . In this case, J 1 = { 1 , 2 , 3 } , J 2 = { 2 , 3 , 4 } , J 3 = { 3 , 4 } , J 4 = { 1 , 2 , 4 } . This system co rr espo nds to the naive information fl ow graph depicted in F igure 4b. A. Informatio n flow graph of a decentralized system W e have seen above that the n aiv e definition of infor- mation flow is no t satisfactory since it depends on the parametriza tion chosen fo r the system, wherea s decentra l- ization is a coord inate-free no tion: our ch oice of co ordina tes to d escribe a system should not affect the knowledge each agent has about the system. W e provide her e a coo rdinate free d efinition of in formatio n flow . The id ea is to let the observation functions h i , define the vertices of the gr aph, and use the vector fields g i to determine the p resence of edge s, Precisely , to a decen tralized con trol system of th e typ e ˙ x = X i u i ( δ i ( µ ); h i ( x )) g i ( x ) we will assign a directed g raph at first, and refine the no tion to obtain a simplicial complex . Recall that a vector field g ( x ) on M acts on functio ns h defined o n M via differentiation. W e write this action as g · h ( x ) . If h ( x ) = [ h 1 , . . . , h k ] is vector-v a lued, we define g · h as g · h = [ g · h 1 , . . . g · h k ] . For example, on R n with c oordin ates ( x 1 , . . . , x n ) , the vector field s g 1 ( x ) = [ g 11 ( x ) , . . . , g 1 n ( x ] and g 2 ( x ) = [ g 21 ( x ) , . . . , g 2 n ( x )] act o n th e function h ( x ) v ia g i ( x ) · h = n X j =1 g ij ∂ ∂ x j h. The L ie bracket of g 1 and g 2 is the vector field [ g 1 , g 2 ]( x ) = ∂ g 2 ∂ x g 1 − ∂ g 1 ∂ x g 2 where ∂ g ∂ x is th e Jacob ian m atrix of g . Definition 8 (Info rmation Flow Graph) . Con sider the decen- tralized co ntr ol system ˙ x = n X i =1 n i X j =1 u ij ( δ i ( µ ); h i ( x )) g ij ( x ) (4) wher e all the functio ns and vector fields involved ar e smooth. W e assign to this system the graph with n vertices h 1 , h 2 , . . . , h n and edges g iven acc or d ing to the fo llowing rules: n i = 1 th er e is an edge fr om h j to h i if g ik ( x ) · h j ( x ) 6 = 0 for any k = 1 . . . n i n i 6 = 1 Let { g i 1 , . . . , g in i } LA be the set of vector fields obta ined by ta king iterated Lie b rac kets of g i 1 , . . . , g in j . Ther e is an edge between fr om h j to h i if g ik ( x ) · h j ( x ) 6 = 0 for any g ik ∈ { g i 1 , . . . , g in i } LA . In wor d s, ther e is an e dge fr om h j to h i if the motion of an agent that uses the observatio n fun ction h i is observable by h j . In the mu lti-agent case, each age nt will often have its own ob servation fun ction, and the ab ove can be r ephrased as sayin g that there is an edge fro m agent i to agent j if agent i can ob serve change s in the state o f ag ent j . Example 1 0. Consider the system on R 4 given by ˙ x 1 = u 1 ( x 1 , x 2 ) ˙ x 2 = u 2 ( x 2 , x 3 ) ˙ x 3 = u 3 ( x 3 , x 4 ) ˙ x 4 = u 4 ( x 4 , x 1 ) W e let x = ( x 1 , x 2 , x 3 , x 4 ) and h 1 ( x ) = ( x 1 , x 2 ) , h 2 ( x ) = ( x 2 , x 3 ) , h 3 ( x ) = ( x 3 , x 4 ) and h 4 ( x ) = ( x 1 , x 4 ) . W e de fine g 1 ( x ) = [1 , 0 , 0 , 0] , g 2 ( x ) = [0 , 1 , 0 , 0] , . . . , g 4 ( x ) = [0 , 0 , 0 , 1] The system o f Equation (5) can thus be written as ˙ x = X i u i ( h i ( x )) g i ( x ) . Accor ding to d efinition 12, the in formation flow graph is given by G = ( V , E ) with V = { h 1 , h 2 , h 3 , h 4 } and E = { ( h 1 , h 2 ) , ( h 2 , h 3 ) , ( h 3 , h 4 ) , ( h 4 , h 1 ) } . The same system expr essed in the z variab les defined in Example 9 is g iven by ˙ z 1 = u 1 ( z 1 − z 2 , z 2 − z 3 ) + u 2 ( z 2 − z 3 , z 3 − z 4 ) + u 3 ( z 3 − z 4 , z 4 ) + u 4 ( z 4 , z 1 − z 2 ) ˙ z 2 = u 2 ( z 2 − z 3 , z 3 − z 4 ) + u 3 ( z 3 − z 4 , z 4 ) + u 4 ( z 4 , z 1 − z 2 ) ˙ z 3 = u 3 ( z 3 − z 4 , z 4 ) + u 4 ( z 4 , z 1 − z 2 ) ˙ z 4 = u 4 ( z 4 , z 1 − z 2 ) . The observation fu nctions a r e ˜ h 1 ( z ) = ( z 1 − z 2 , z 2 − z 3 ) , ˜ h 2 ( z ) = ( z 2 − z 3 , z 3 − z 4 ) , ˜ h 3 ( z ) = ( z 3 − z 4 , z 4 ) , ˜ h 4 ( z ) = ( z 4 , z 1 − z 2 ) . The contr ol vector fi elds become ˜ g 1 ( z ) = [1 , 0 , 0 , 0] , ˜ g 2 ( z ) = [1 , 1 , 0 , 0 ] , ˜ g 3 ( z ) = [1 , 1 , 1 , 0 ] and ˜ g 4 ( z ) = [1 , 1 , 1 , 1] . W e ca n n ow write ˙ z = X i u i ( ˜ h i ( z )) ˜ g i ( z ) . The in formation flow graph associa ted to this syst em has four vertices ˜ h 1 , ˜ h 2 , ˜ h 3 , ˜ h 4 W e ha ve th e following r elations ˜ g 1 · ˜ h 2 = [ ∂ z 1 ( z 2 − z 3 ) , ∂ z 1 ( z 3 − z 4 ) = [0 , 0] ˜ g 1 · ˜ h 3 = [0 , 0]; ˜ g 1 · ˜ h 4 = [0 , 1] ˜ g 2 · ˜ h 1 = [0 , 1]; ˜ g 2 · ˜ h 3 = [0 , 0] ˜ g 2 · ˜ h 4 = [0 , 0]; ˜ g 3 · ˜ h 1 = [0 , 0] ˜ g 3 · ˜ h 2 = [0 , 1]; ˜ g 3 · ˜ h 4 = [0 , 0] ˜ g 4 · ˜ h 1 = [0 , 0]; ˜ g 4 · ˜ h 2 = [0 , 0] ˜ g 4 · ˜ h 3 = [0 , 1] These r elations yield th e sam e information flo w g raph as above. B. Decentralized systems and Whitney complex Definition 12 in the previous sectio n attaches an infor ma- tion flow grap h to a d ecentralized system in a coord inate free manner . The salient po int was that the ob servation function s h i ( x ) provide a natura l set of vertices for the g raph. Consider the triangu lar formation of Figu re 5a. The main source of difficulty in the con trol of th is fo rmation comes from the fact that th e m otion of x i depend s on the motion of x i +1 (taken modulo 3 ): if x 2 moves, x 1 has to adjust itself, which forces x 3 to move which in tu rn provokes a motion o f x 2 . W e call this a no ntrivial lo op of in formation . Now con sider th e trian gular formation with bidirectiona l edges. The above me ntioned loop of informatio n still exists, but its effect on the dyn amics is d iluted due to the fact that the commu nication g oes both ways between the agents. In fact, th ere is all-to-all co mmun ication between agents in this fo rmation, and the system is thus equ iv alen t to a centralized one, wh ere each agent imp lements locally a copy of a cen tralized con troller . W e call this informa tion loop trivial. W e devote the remainder of this section to putting this notion of triviality on a firm mathematical fo oting. In o rder to do so, we need som e co ncepts fro m algebr aic topolog y . Th e r ole of ho mologic al a lgebra and alg ebraic topolog y in contro l th eory and applied scien ces has b een recogn ized in many different contexts such as feedb ack stabilization, co mputer graph ics, sensor networks or data analysis [17], [1 8], [19], [2 0]. W e show here h ow related ideas naturally appear in th e d efinition o f d ecentralized systems. W e start with som e graph theoretic definitions. W e r ecall here that the infor mation flow graph is in gener al a mixed graph (i. e. containing bo th directed and undirected edg es). W e say that G = ( V , E ) is an und ir ected complete graph if E = { ( v i , v j ) s.t. v i , v j ∈ V } . In words, G contains all possible undire cted edges o n its vertices. If G = ( V , E ) is a graph , we call G ′ = ( V ′ , E ′ ) the subgr aph of G generated by V ′ ⊂ V when E ′ ⊂ E is the set of edges of E which start and end at vertices in V ′ : E ′ = { ( v i , v j ) ∈ E for all v i , v j ∈ V ′ } . A subgr aph G ′ of G is an u ndir ected clique if it is an undirected com plete graph . W e can n ow give a coordin ate indepen dent definition of dec entralized system s: Definition 9 ( Decentralized system) . A system o f type of Equation (8) is centralized if its associated information flow is a n u ndir ected complete graph. Otherwise, it is decentral- ized. W e now address th e fact that some informatio n loops ar e trivial, as describ ed in the begin ning of this section. A p ath of length k in a g raph G = ( V , E ) is a n ordered list of vertices v 1 , . . . , v k , without repetitions except possibly for v 1 and v k , such that ( v i , v i +1 ) ∈ E for i = 1 . . . , k − 1 . A path is closed or a loop if v 1 = v k . Definition 1 0 (I nform ation loop) . A nontrivial in formation loop in a decentralized system with information fl ow g raph G = ( V , E ) is a closed pa th ( v 1 , . . . , v k ) such th at the subgraph G ′ generated by V ′ = { v 1 , . . . , v k } is not an undirected c lique. This definition takes in to account the fact th at when a graph is fully conn ected, even th ough loops will exist, the ir presence h as no effect o n th e dynamics o f the system. The definition of inf ormation lo op points towards the use of techniques f rom homolo gical algebra to handle the informa tion flow g raph. W e define here a combin atorial object, called simplicial complex, which allows us to make a con nection between the structure of decentralized systems and algebraic top ology . A k-simplex is determined by k+1 vertices; we use the usual notation [ x 1 , x 2 , . . . , x k +1 ] for the k- simplex with vertices x 1 , . . . , x k +1 . A k-simp lex has k+1 facets which are (k-1)- simplices, they are given by [ x 2 , . . . , x k ] , [ x 1 , x 3 , . . . , x k ] , . . . , [ x 1 , . . . , x k − 1 ] . Definition 11. An abstrac t simplicial co mplex S is a combi- natorial object co nsisting of a set of simplices such th at a ny facet o f a simp lex s ∈ S is also in S . The k- skeleton o f a simplicial co mplex is the set of simplices of d imension k o r less. W e have the f ollowing definition: Definition 12 (Info rmation Flow Com plex) . Consider the decentralized contr ol system ˙ x = n X i =1 n i X j =1 u ij ( δ i ( µ ); h i ( x )) g ij ( x ) wher e a ll the fun ctions and vector fie lds in volved ar e smo oth. W e a ssign to this system the simplicia l co mplex with n vertices h 1 , h 2 , . . . , h n and facets given acco r din g to th e following: 1) Ther e is an edge between x i and x j if g j k ( x ) · h i ( x ) 6 = 0 for any k = 1 . . . n j 2) There is a k -simplex with v ertices x 1 , . . . , x k if g j k · h i 6 = 0 for any g j k ∈ { g j 1 , . . . , g j n j } LA , for all i, j = 1 ..k h 1 h 2 h 3 (a) h 1 h 2 h 3 h 4 (b) Fig. 5 : In ( a ) , we rep resent the in formatio n flow com plex o f system (5), which exhibits a no ntrivial information loop. In ( b ) , we represent the info rmation flo w complex of system (6); the shaded region de picts a 2-simplex in the complex. The informa tion loop between 1 , 2 and 3 is tri vial in th is case. The simplicial co mplex defin ed a bove is sometimes called a Whitney co mplex or flag comp lex in the literatu re. Due to space c onstraints, an d the amou nt of back grou nd nece ssary to analyze such objects any further, m ost notably via their cohomo logy gro ups, we leave the stud y of th e inf ormation flow comp lex to fu ture work. Example 1 1. Consider the system    ˙ x 1 = u 1 ( x 1 , x 2 ) ˙ x 2 = u 2 ( x 2 , x 3 ) ˙ x 3 = u 3 ( x 3 , x 1 ) (5) Using th e notation in tr odu ced above, we ha ve h 1 ( x ) = ( x 1 , x 2 ) , h 2 ( x ) = ( x 2 , x 3 ) , h 3 ( x ) = ( x 3 , x 1 ) and g i ( x ) = e i . W e thus hav e g 1 · h 2 = [0 , 0]; g 1 · h 3 = [0 , 1] g 2 · h 1 = [0 , 1]; g 2 · h 3 = [0 , 0] g 3 · h 1 = [0 , 0]; g 3 · h 2 = [0 , 1] W e thu s associate the information comp lex S = { h 1 , h 2 , h 3 , [ h 1 , h 2 ] , [ h 2 , h 3 ] , [ h 3 , h 1 ] } . 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