Technical Notes and Correspondence

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 8, AUGUST Technical Note and Correpondence Information Flow and It Relation to Stability of the Motion of Vehicle in a Rigid Formation Sai Krihna Yadlapalli, Swaroop Darbha, and K. R. Rajagopal Abtract In thi note, we conider the effect of information flow on the propagation of error in pacing in a collection of vehicle trying to maintain a rigid formation during tranlational maneuver. The motion of each vehicle i decribed uing a linear time-invariant (LTI) ytem. We conider undirected and connected information flow graph, and aume that each vehicle can communicate with a maximum of vehicle, where may vary with the ize of the collection. We conider tranlational maneuver of a reference vehicle, where it teady tate velocity i different from it initial velocity. In the abence of any diturbing force acting on the vehicle during the maneuver, it i deired that the collection be controlled in uch a way that it motion aymptotically reemble that of a rigid body. In the preence of bounded diturbing force acting on the vehicle, it i deired that the maximum deviation of the motion of the collection from that of a rigid body be bounded and be independent of the ize of the collection. We conider a decentralized feedback control cheme, where the controller of each vehicle take into account the aggregate error in poition and velocity from the vehicle with which it i in direct communication. We aume that all vehicle tart at their repective deired poition and velocitie. Since the diplacement of every vehicle at the end of the maneuver of the reference vehicle mut be the ame, we how that the loop tranfer function mut have at leat two pole at the origin. We then how that if the loop tranfer function ha three or more pole at the origin, then the motion of the collection i untable, that i, it deviation from the rigid body motion i arbitrarily large, if the ize of the formation i ufficiently large. If i the number of pole of the tranfer function relating the poition of a vehicle with it control input, we how that if ( ( ) ) 0 a, then there i a low frequency inuoidal diturbance of at mot unit amplitude acting on each vehicle uch that the maximum error in pacing repone increae at leat a ((( ( )) )). 1 A conequence of the reult preented in thi note i that the maximum error in pacing and velocity of any vehicle can be made inenitive to the ize of the collection only if there i at leat one vehicle in the collection that communicate with at leat ( ) other vehicle in the collection. Index Term Autonomou vehicle, decentralized control, formation, multivehicle ytem, calability. I. INTRODUCTION The development of technologie related to unmanned vehicle ha received coniderable attention in the literature [1] [6]. Recently, the problem of controlling the motion of unmanned vehicle to reemble that of a rigid body ha attracted ignificant interet [1] [6] and it i to thi problem that thi note i devoted. In thi note, we treat an unmanned vehicle (or imply, a vehicle) a a point ma. It i clear that if there are n vehicle in a three dimenional formation, there are 3n Manucript received November 10, 2004; revied November 15, Recommended by Aociate Editor F. Bullo. Thi work wa upported by the National Science Foundation under Award The author are with the Department of Mechanical Engineering, Texa A&M Univerity, College Station, TX USA ( dwaroop@mengr. tamu.edu). Digital Object Identifier /TAC A function p(n) i (q(n)) if there exit a nonzero contant c >0 and a N uch that jp(n)j cjq(n)j for all n > N. degree of freedom. Conequently, if the motion of the formation were to reemble that of a rigid body, 3n06 degree of it freedom mut be arreted for a three dimenional formation. Contraining the degree of freedom can be achieved by mean of virtual rigid link (or imply virtual link or more imply, link) between vehicle. The maintenance of uch a link between two vehicle require the relative poition and velocity information between the two concerned vehicle and an actuation mechanim to provide the relevant control force to maintain the pacing between the vehicle. Since the maintenance of virtual link between vehicle require either ening or communication between concerned vehicle, it i natural to ak, from the viewpoint of cot and implementation, which are the appropriate link that mut be maintained. In thi note, we are concerned with related but different quetion how many link hould be attached to each vehicle in the formation and what other contraint come into play on the tructure of the controller? We conider a homogeneou collection of vehicle and equip each vehicle with the ame controller. The controller aggregate the pacing error of all vehicle it i in communication with (or can ene) and yntheize a control action. The purpoe of the controller i to regulate the intervehicular pacing between any pair of vehicle that maintain a link. To quantify the deviation of the motion of the vehicle, we will conider tranlational maneuver by a reference vehicle in the collection. Since a rigid body motion i deired by the formation, the deired poition of any vehicle i automatically pecified from the poition of the reference vehicle. We conider a pacing error of a vehicle to be the deviation in the poition of a vehicle from it deired poition. It i ueful to have the ame controller on every vehicle from the viewpoint of implementation each vehicle doe not have to know the ize of the formation or it index in the collection. One can therefore be unconcerned with the iue of calability of the ize of the formation if equipping each vehicle with an identical controller i poible. Similarly, if each vehicle were to maintain only a few communication or ening link irrepective of the ize of the formation, the communication/ening requirement cale well and are eay to implement. We will how in thi note that for certain clae of controller and information flow graph, the calability of controller i at odd with calability of communication/ening requirement. For thi purpoe, we tudy the propagation of pacing error of vehicle in the formation when they are ubject to diturbing force of bounded magnitude (without any lo of generality, we aume the bound to be unity). The propagation of error depend upon the virtual link that are maintained. The graph pecifying the pair of vehicle maintaining a link i referred to a an information flow graph throughout thi note. We call a controller calable with repect to an information flow graph if the following hold. 1) In the abence of any diturbing force or initial error in the poition of the vehicle, the diplacement of all vehicle in the formation, in repone to any maneuver of the formation where the teady tate peed i different from it initial peed, i the ame aymptotically. Thi enure that the formation retain rigidity to within a perturbation that vanihe aymptotically at leat for thi cla of maneuver. 2) In the preence of diturbing force, the reulting pacing error (which meaure the deviation of the motion of the formation from that of a rigid body) are bounded by a contant independent of the ize of the collection /$ IEEE

2 1316 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 8, AUGUST 2006 We will refer to any controlled formation a an approximately rigid formation or imply, by abue of notation, a rigid formation, if it atifie the previou two condition for ome controller. In thi note, we conider connected and undirected information flow graph and we how that ome vehicle in the formation mut maintain (n) link (eentially, it i a lower bound on the degree of the undirected information flow graph) o that ome rational, proper controller, C() i calable; further, if P () i the tranfer function of the vehicle, we how that the loop tranfer function P ()C() mut have exactly two pole at the origin. Should the loop tranfer function have more than two pole at the origin, for a ufficiently large formation, the motion of the formation will be untable. 2) The underlying information flow graph i connected and undirected. A mechanical analog of thi aumption i the following: If link were to be connected between appropriate vehicle decribed by the information flow graph, and if the link were not rigid but behave like linear pring, then we obtain a pring-ma ytem with exactly one rigid body mode; however, if the link were rigid, then we obtain a rigid body. Let S i be the et of link incident on (or maintained by) the ith vehicle. If j 2 S i, it implie that there i a link between vehicle i and j. Clearly, j 2 S i =) i 2 S j. 3) Let L ij be the vector of deired ditance of the ith vehicle from the jth vehicle. Clearly, L ij = L i 0 L j. The controller i of the form II. PROBLEM FORMULATION Throughout thi note, we will ue the following notation: F () i the Laplace tranformation of f (t) and jsj i the ize of the et S.We will ue A B to repreent the Kronecker product of A and B. For any two vector x, y 2< n, we will ue hx; yi to repreent the inner product between them; here, we will ue the tandard inner product n =1 x y, where x and y are the th component of x and y, repectively. The term det A will repreent the determinant of a matrix A and I n will denote an identity matrix of dimenion n. We will ue p1, p2, and p3 to repreent the firt, econd and third column of I3 and will repreent a n dimenional vector whoe component are all 1 by 1 n. We will conider a formation coniting of n vehicle. Let x i (t) 2 < 3 denote the poition of the ith vehicle in the formation at time t. Without any lo of generality, we will aume that the firt vehicle i a reference vehicle. Let L i 2< 3 be the vector of deired ditance of the ith vehicle from the reference vehicle. We will refer to the error in pacing of the ith vehicle a U i () =C() where C() i a rational tranfer function matrix of the controller, and the term in the parenthei on the right-hand ide of the previou equation i the poition error in the ith vehicle with repect to the jth vehicle. Our motivation for thi tructure tem from the need to yntheize a control action baed on the information available to a vehicle. We alo retrict C() to be uch that the loop tranfer function ((I3 + G)) 01 H()C() to be proper. In eence, the ith vehicle aggregate the error in it pacing with repect to all the vehicle with which it i maintaining a link and thi aggregate error i proceed by the controller to produce a control force for the ith vehicle. 4) We will aume that the initial error in pacing and velocity are zero, i.e., 8i 2, e i(0) = 0, _e i(0) = 0. With the aumed tructure of the controller, the control force acting on the ith vehicle i (2) e i (t) :=x i (t) 0 x1(t) 0 L i 8i 2: We note that the ith vehicle may not have the information about the error in pacing with repect to the reference vehicle. Thi may be due to the lack of an onboard enor which meaure the relative poition of the ith vehicle with repect to the reference vehicle or due to lack of communication between the ith vehicle and the reference vehicle. We aume the following. 1) The motion x i (t) can be adequately decribed uing the following differential equation: x i = 0G _x i + f i 0 d i (1) where G 0 i a diagonal matrix coniting of damping coefficient, f i (t) 2 < 3 i the force input from an actuator and d i (t) 2< 3 i the diturbance acting on the ith vehicle. We will aume that f i i related to the control input through a tranfer function of the formf i () =H()U i (), where u i (t) i the control input. We will how later in thi ection that there i no lo of generality in auming that H() =I3. The aumption to treat the controlled motion of a vehicle a that of a point ma i not unreaonable if the acceleration of every degree of freedom of a vehicle i controllable (a i the cae in an aircraft or in the longitudinal motion of a ground vehicle) through an independent input, thi model can be obtained after feedback linearization. Although we only deal with three-dimenional formation, the reult preented in thi note are alo applicable to formation in other dimenion, where the motion in each dimenion can be independently controlled. F i () =H()C() : (3) One can eaily aborb the tranfer function H() into the controller tranfer function and treat f i (t) a a control input. Without any lo of generality in analyzing the tability of controlled motion of vehicle, we will therefore rename our control input f i(t) a u i(t) (thereby treating H() I3) and the controller C() now relate how the control force, u i(t), change with repect to aggregate error in pacing of the ith vehicle. An alternative to the controller tructure provided in (2) i aumed by Fax and Murray [5] U i () =C() 1 js ij : (4) The reult preented in thi note alo hold for thi type of controller tructure and we will outline how our reult apply to thi tructure at appropriate place throughout thi note. Information flow graph are typically decribed uing a Laplacian (or in the cae of Fax and Murray controller, a combinatorial Laplacian) matrix. A Laplacian matrix for an undirected information flow graph reemble that of a tiffne matrix in a pring ma ytem. If there i a link between node i and j, there i a pring of unit tiffne connecting point mae i and j in the mechanical analog. The reulting tiffne matrix and the Laplacian matrix of the information flow graph have identical entrie. We will ue K to denote the Laplacian and clearly, for all i 6= j and 1 i, j n, K ij = 01 and K ii = Kij. Let j6=i

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 8, AUGUST P () :=((I 3 + G)) 01 refer to the tranfer function of a vehicle and let X 1 () :=X 1 () 0 x1(0) 0 P ()_x 1 (0): (5) We conider two cenario. 1) We allow no diturbing force on any vehicle. The reference vehicle make a maneuver where it attain a teady-tate velocity v f different from it initial velocity. Thi mean that lim!0 2 X 1 () =v, for ome v 2< 3. 2) We allow diturbing force (of magnitude at mot one unit) to act on each vehicle. The objective of a calable controller deign are two fold: In the firt cenario, the pacing error mut decay aymptotically, i.e., lim t!1 e i(t) =0for all i 2. In the econd cenario, it i deired that there be a M > 0 that i independent of the ize of the formation uch that max 1infe i(t); _e i(t)g M for every ize n of the formation and for all t 0. The econd objective i related to colliion avoidance. We are intereted in determining neceary condition on the degree of the information flow graph and on the tructure of the controller C() o that the two objective mentioned previouly are met. III. MAIN RESULTS Before we proceed to preent our reult, we will derive the equation governing the evolution of pacing error and point out ome mathematical fact which are neceary for deriving the reult. Due to Aumption 4, one may write the pacing error for the ith vehicle a E i () =X i () 0 X 1 () 0 Li (6) = P ()[U i() 0 D i()] 0 X 1(): (7) Uing the fact that x i(t) 0 x j(t) 0 L ij mut equal (x i(t) 0 x 1(t) 0 L i) 0 (x j(t) 0 x 1(t) 0 L j(t)), and uing the (2) and (6), we get for all i 2: (I 3 + P ()C()jS i j) E i () = P ()C() E j () 0 P ()D i () 0 X 1 (): (8) Let K 1 be the principal minor of the Laplacian matrix K by removing the firt row and column (they correpond to the reference vehicle). We will contruct a vector, e(t) 2< 3(n01), of error by tacking e 2(t);...;e n(t). Similarly, we will contruct a vector d(t) 2< 3(n01) of diturbance. We will contruct another vector x(t) 2< 3(n01) by tacking (n 0 1) copie of x 1 (t). One may now write the aforementioned et of governing equation for pacing error compactly a [I 3n03 + K 1 (P ()C())] E() =0 (I n01 P ()) D()0 X(): (9) Let diag(k 1 ) be a diagonal matrix whoe diagonal entrie are the ame a thoe of K 1. With the control tructure of Fax and Murray given by (4), the following equation hold: (I 3 + P ()C())E i () = P ()C() E j () 0 P ()D i () 0 X js 1 (): (10) ij I 3n03 + (diag(k 1)) 01 K 1 (P ()C()) E() = 0 (I n01 P ()) D() 0 X(): (11) We oberve from the ymmetry of K and the Gerchgorin dic theorem that K i poitive emidefinite [7]. Uing the Cauchy interlacing theorem [7] and the fact that the information flow graph i connected, we note that K 1 i poitive definite. Let v 2< n01 be an eigenvector correponding to an eigenvalue of K 1. Let p 1 ;p 2 ;p 3 2< 3 be the firt, econd, and third column of the identity matrix and let v be an eigenvector correponding to the mallet eigenvalue of K 1. Let A be any matrix. Then, one can how that (I n01 A)(v I 3 )= (v I 3 )A and (K 1 A)[v p 1 v p 2 v p 3 ]= [v p 1 v p 2 v p 3 ] A: (12) The previou equation alo indicate that the pace panned by vp i, i =1; 2; 3 i invariant under K 1 A. A. Analyi of the Evolution of Spacing Error We will tart with an analyi of the firt control objective. Since the diturbing force are abent, one may rewrite equation (9) a E() = 0 [I 3n03 + K 1 (P ()C())] 01 2 X() (13) ) lim E() = 0 lim (K1 (P ()C()))01!0!0 2 1 n01 v : (14) Lemma 3.1: A neceary condition for lim t!1 e(t) =0for every poible final velocity v i that the loop tranfer function matrix P ()C() mut be expreible a L()= k for ome k 2 and for ome tranfer function matrix L() that doe not have any pole at the origin and L(0) i noningular. Proof: We oberve that k 1; otherwie, there i a cancelation of a pole of the open loop ytem at 0 and a zero of the controller at 0. Thi would imply that the controlled motion i not aymptotically table. The characteritic equation det(i 3n03 + K 1 (P ()C())) = 0 may be rewritten a det( k I 3n03 + K 1 L()) = 0. IfL(0) were ingular, 0 will be a root of the characteritic equation and the controlled motion will not be aymptotically table. Clearly, the teady-tate error i lim!0((k 01 1 )(P ()C())01 )(1 n01v )= lim!0(k n01) (P ()C()) 01 v.ifk =1, the teady-tate value of the error e i given by (K ) (L(0) 01 v ). Thi cannot be zero for all poible value of v unle L(0) 01 =0; but thi i impoible, ince L() doe not have any pole at 0. We will prove the main reult concerning the tability of controlled motion of vehicle in a formation by uing the mechanical analogy between the Laplacian of the information flow graph and the tiffne matrix. For ytem that do not have a rigid body mode, a route to intability in tructural mechanic i that the mallet eigenvalue of the tiffne matrix goe to zero. For the purpoe of analyzing the propagation of error in the preence of a diturbance in the econd cenario, we will ground the reference vehicle to zero, i.e., we et X 1 () 0 (or equivalently X() 0 with our aumption). The reulting ytem i analogou to a pring-ma ytem with no rigid body mode. The mechanical analogy indicate the following line of proof, which we adopt here. 1) The mallet eigenvalue of K 1 goe to zero a n!1, when ome vehicle in the formation doe not communicate with (n) other vehicle. 2) Let v be the correponding eigenvector and be caled uch that kvk 1 =1. The analogy indicate the propagation of the error

4 1318 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 8, AUGUST 2006 in pacing, e(t) when the diturbance acting on the vehicle, d(t) are of the form poitive 3 uch that one root of the following equation ha a poitive real part for all < 3 : d(t) =0 in(wt)(v I 3 ) d x d y det [I 3 + P ()C()] = 0: w i the firt natural frequency or any frequency ufficiently cloe to it. The degree q of an information flow graph i the maximum number of link maintained by any vehicle in the formation. We will let q vary with the ize n of the formation. Clearly, q(n) n 0 1 and lim (q(n)=n) 1. The following lemma provide an upper bound on the mallet eigenvalue a a function of q(n) and n. Lemma 3.2: q(n) n 0 1 and if lim q(n)=n =0, then! 0 a n!1. Proof: The ith component of K 1 1 n01 equal zero, if (i+1) 62 S 1 and equal 1, otherwie. An application of Rayleigh inequality [7] yield h1n01;k11n01i h1 n01; 1 n01i d z = js1j n 0 1 q(n) n 0 1 : (15) Since 0, it follow that! 0 a n! 1 whenever lim q(n)=n =0. The upper bound on the mallet eigenvalue can be tightened if the tructure of the information flow graph i pecified a priori; for example, if the ditribution of vehicle whoe length of the hortet path to the reference vehicle in the information flow graph i uniform, then the mallet eigenvalue can be hown to be upper bounded by C 3 =n 2 for ome poitive contant C 3 [10]. We remark that, in the cae of controller given by Fax et al., one mut conider the generalized eigenvalue problem, K 1 v = diag(k 1 )v; ince K 1 i poitive definite and the diagonal entrie of K 1 are alo poitive, all the generalized eigenvalue are poitive and there exit a full et of generalized eigenvector which are orthogonal in the following ene: If v i and v j are two ditinct generalized eigenvector, hv i ; diag(k 1)v j i =0and kv i k 1 =1. Oberving that the diagonal entrie of K 1 are alway at leat 1, through a traightforward application of Rayleigh inequality, the mallet generalized eigenvalue alo atifie the inequality (q(n)=n 0 1). Eentially thi reult implie that ome vehicle in the collection mut maintain a et of link whoe ize i at leat ome non-trivial fraction c 0 2 (0; 1] of the ize of the formation if were required not to decay to zero. In the ret of thi note, we will focu on the cae when! 0. The firt reult indicate the tructure of the loop tranfer function P ()C() that i required for tability of motion of the formation. Lemma 3.3: If lim q(n)=n =0and the loop tranfer function i expreible a L()= k for ome k 3, then there i a critical ize N 3 of the formation uch that for all n>n 3, the motion of the formation of vehicle i untable. Proof: By Lemma 2,! 0 a n!1. Given any 3 > 0, there exit a N 3 > 0 uch that < 3 for all n>n 3. Let 1 = and 1;...; n01 be the eigenvalue of K 1. From the propertie of Kronecker product [or uing (12)], the determinant of [I 3n03 + K 1 (P ()C())] can be expreed a the product of determinant of [I 3 + P ()C() i], i = 1; 2;...;n0 1. The root of det[i 3n03 + K 1 P ()C()] are the union of the root of det[i 3 + P ()C() i ], i =1; 2;...;n0 1. Therefore, it uffice to how that there exit a We will therefore concentrate on the root of 1() := det[i 3 + P ()C()]. By Lemma 1, we will require P ()C() to be of the form L()= k for ome rational tranfer function matrix L() o that L(0) i noningular and L() doe not have pole at 0. Since >0 (by the poitive definitene of K 1 ), let := 1=k. The characteritic equation may be expreed a det[ k I 3 + k L()] = 0: Clearly, when =0, there are 3k root of 1() at 0. We are intereted in the movement of thee root for ufficiently mall poitive value of.for thi purpoe, we will employ regular perturbation technique [8]; let the root at origin move a = 1 + O( 2 ), where the direction of a root 1 need to be determined. Evaluating 1() and etting to zero the lowet order term in (in thi cae, O( k ) term), we get det k 1I 3 + L(0) =0: (16) If i, i =1; 2; 3 are the eigenvalue (not necearily real) of L(0), then the lat equation may be rewritten a k k k =0: (17) Since L(0) cannot be ingular, it follow that if k 3, at leat one root of the previou polynomial equation for 1 ha a root with poitive real part, implying that one root of the characteritic polynomial ha a poitive real part if k 3. Remark 3.1: Lemma 1 and 3 imply that the loop tranfer function mut only be expreible a L()= 2. In thi cae, if lim q(n)=n =0, then the eigenvalue 1, 2, 3 mut be poitive; otherwie, following the proof of Lemma 3, one can argue that for a ufficiently large formation, the motion will be untable. Remark 3.2: Uing the imilar analyi a in the aforementioned lemma, but with maximum eigenvalue, it can be hown that for if lim q(n)=n = 0 and the loop tranfer function i expreible a L()= 2 and L() ha zeroe in the right-half plane, then there i a critical ize of a formation N ~ uch that for all n> N ~, the motion of the formation of vehicle i untable. Remark 3.3: We oberve again that the eigenvalue and eigenvector of diag(k 1) 01 K 1 are the ame a the generalized eigenvalue,, and the correponding eigenvector v, given by K 1v = diag(k 1)v. Since the root of the characteritic equation can be expreed a the union of the root of det(i 3 + ip ()C()) = 0, where i i the ith generalized eigenvalue, the previou theorem alo hold for the controller of Fax and Murray [5]. Now that we have obtained the neceary condition on the tructure of the loop tranfer function P ()C() that i required for tability of motion of the formation, we will hift our focu to the analyi of the propagation of error for bounded diturbing force acting on every vehicle. Theorem 3.1: Conider a formation of n vehicle. Suppoe lim q(n)=n = 0 and the loop tranfer function i expreible a L()= 2 and L(0) having real and poitive eigenvalue. Then, there i a inuoidal diturbance acting on each vehicle of the ame frequency and at mot unit in magnitude, uch that the error in the pacing i of ((n=q(n)) (l+1)=2 ), where l i the mallet poitive integer uch that lim!0 l P () i bounded. Proof: We will conider the error propagation (9) with X() 0. We will conider a diturbance d(t) which may be expreed for ome

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 8, AUGUST real d x, d y and d z a d(t) =d xv p 1 + d yv p 2 + d zv p 3. Let T de () =(I 3 + P ()C()) 01 P (). Becaue of (12) and the fact that (I n01 P ())(v I 3 ) = (v I 3 )P (), one may expre e(t) = e xv p 1 + e yv p 2 + e zv p 2 for ome real e x, e y, and e z; moreover ) E x () E y() E z() E x(jw) E y(jw) E z (jw) =(I 3 + P ()C()) 01 P () 2 T D x() D y () D z () () : (18) = I 3 0 L(jw) w 2 01 P (jw) 2 D x (jw) D y(jw) D z(jw) : (19) Conider a inuoidal diturbance force acting on each vehicle in each direction at the following frequency: w = i(0) where i(0) i an eigenvalue of L(0). At that frequency, the amplitude and phae hift are given by (I 3 0 (L(jw)= i(0))) 01 P(jw), which may be expreed a I 3 0 L(jw) i (0) 01 (jw) l 1 P (jw) (jw) l where l i the mallet poitive integer uch that lim!0 l P () i bounded. It hould be noted that l =1when G i poitive definite and l =2, otherwie. Since lim w!0(i 3 0(L(jw)= i (0)))i ingular, w p for ome p 1 i a factor of the det(i 3 0 (L(j!)= i (0))). Hence, we may rewrite, for all ufficiently mall w, (I 30(L(jw)= i(0))) 01 a (1=w p ) L(jw), ~ where L(0) ~ i bounded and not identically zero. Therefore, the frequency repone of the tranfer function i given by ~L(jw)(0j) l (1=w p+l ). Since i(0) = w 2, the amplitude of error, (e x (t);e y (t);e z (t)) increae a p (1= p+l ) for ome p 1 for low frequency diturbance. Therefore, the maximum ingular value of the tranfer function, T de () i at leat p (1= p+l ). When (d x ;d y ;d z ) i choen along the appropriate ingular direction, the amplitude of at p leat one entry in (e x(t);e y(t);e z(t)) at leat increae a (1= l+1 ). By Lemma 2 we have, (q(n)=n 0 1). Therefore, the error in the pacing increae a (( n=q(n)) l+1 ) for ufficiently large formation. Remark 3.4: The aforementioned theorem alo hold for the controller of Fax and Murray [5]. Since a full et of eigenvector exit for diag(k 1 ) 01 K 1, let v be the eigenvector correponding to the mallet eigenvalue of diag(k 1 ) 01 K 1. Then, one ha I 3n03+ (diag(k 1 ) 01 K 1 (P ()C()) [vp 1 vp 2 vp 3 ] =[vp 1vp 2vp 3][I 3 +P ()C()] : (20) A a reult, if the diturbance d = d x v p 1 + d y v p 2 + d z v p 3, then the error i of the form e = e xv p 1 + e yv p 2 + e zv p 3 and further, we have E x() E y () E z () =(I 3 + P ()C()) 01 P () D x() D y () D z () : (21) From the proof of the previou theorem, the ret of the proof of thi remark follow. IV. CONCLUSION In thi note, we introduced the notion of calability of controller and howed that the degree of the underlying information flow graph of a formation mut change with it ize n a (n) for ome controller to be calable. If a controller i calable, it mean that one can equip vehicle with the ame controller and doe not have to worry about linky-type of error propagation which render the controlled motion of formation to deviate ignificantly from the rigid body motion. The reult in thi note point to the tradeoff between calability of communication and calability of controller, i.e., at leat one vehicle in the collection mut maintain a ignificant number of communication link for ome calable controller to exit. In automated highway ytem, the information flow graph of a platoon wa acyclic and the lead vehicle maintained n 0 1 link, n being the ize of the platoon [1]. A imilar information flow graph wa adopted for higher dimenional formation in [4]. Baed on our experience [9], we believe that there mut be (n) communication link maintained by ome vehicle in the platoon for ome control algorithm to be calable and the reult in thi note point in thi direction for undirected information flow graph. We have hown in thi note that the correponding tructure of the controller, C(), mut be uch that the loop tranfer function mut be expreible a a proper rational function L()= 2 and that L(0) mut have poitive eigenvalue. Lemma 2 and Theorem 1 generalize [4, Th. 2.3]. ACKNOWLEDGMENT The author would like to thank the anonymou referee for their meticulou review and for pointing them to trengthening q to be of (n) for the calability of controller. They thank Prof. S. P. Bhattacharyya and Mr. S. Coimbatore for their comment. They alo thank Dr. P. Seiler for hi comment. REFERENCES [1] D. Swaroop and J. K. Hedrick, String tability of interconnected ytem, IEEE Tran. Autom. Control, vol. 41, no. 3, pp , Mar [2] M. Khatir and E. J. Davion, Bounded tability and eventual tring tability of a large platoon of vehicle uing non-identical controller, in Proc IEEE Control and Deciion Conf., Paradie Iland, Bahama, Dec. 2004, pp [3] T. Eren, B. D. O. Anderon, A. S. More, W. Whiteley, and P. N. Belhumeur, Operation on rigid formation of autonomou agent, Commun. Inform. Syt., pp , Sep [4] P. J. Seiler, Coordinated control of unmanned aerial vehicle, Ph.D. diertation, Dept. Mech. Eng., Univ. California, Berkeley, CA, [5] J. A. Fax and R. M. Murray, Information flow and cooperative control of vehicle formation, in Proc. IFAC World Congr., Barcelona, Spain, Jul. 2002, pp [6] J. D. Wolfe, D. F. Chichka, and J. L. Speyer, Decentralized controller for unmanned aerial vehicle formation flight AIAA Paper , Jul [7] G. H. Golub and C. F. Van Loan, Matrix Computation, 2nd ed. Baltimore, MD: John Hopkin Univ. Pre, [8] M. H. Holme, Introduction to perturbation method, in Text in Applied Mathematic. New York: Springer-Verlag, [9] D. Swaroop, String tability of interconnected ytem: An application to platooning in automated highway ytem, Ph.D. diertation, Univ. California, Berkeley, MD, [10] Y. Sai Krihna, Information flow and it relation to the tability of controlled motion of vehicle in a rigid formation, M.S. thei, Dept. Mech. Eng., Texa A&M Univ., College Station, TX, May 2005.