Scalability in heterogeneous vehicle platoonsIoannis Lestas and Glenn Vinnicombe

Department of EngineeringUniversity of CambridgeCambridge CB21TQ, UKicl20,gv@eng.cam.ac.uk

Abstract It is known that vehicle platoons exhibit stringinstability when each vehicle tries to maintain a fixed dis-tance from its predecessor. This can be avoided if sufficientlystrong coupling with the leader is employed. If instead eachvehicle tracks the average distance form its neighbours, theinterconnection can still be ill-conditioned in the sense that theresponse to disturbances is not uniformly bounded with thesize of the platoon. We show in the paper that in a symmetricbidirectional scheme, arbitrarily weak coupling with the leadercan make the platoon scalable. In addition, we show thatdespite the additional feedback in a bidirectional control law,the symmetry of the information flow enables the derivation oflocal conditions which, if satisfied, guarantee that an arbitrarilylong heterogeneous interconnection is robustly stable.

I. INTRODUCTION

The problem considered in the paper is the control ofa platoon of vehicles such that there is a constant spacingbetween consecutive vehicles and with all following a leadervehicle that moves independently. As a result of applicationsin automated highway systems the problem has receivedconsiderable attention by the control community. A majorcomplication, however, in the analysis comes from the factthat on the one hand the control law needs to be decentralizedand at the same time vehicle/controller dynamics containa double integrator. The latter can lead to ill-conditionedbehaviours when issues of scalability become important. Forexample, in the simplest decentralized scheme where eachvehicle tracks a certain distance from its predecessor, it hasbeen shown in [1] that for any linear control law disturbancesare amplified as they propagate along the vehicle string.This follows from a Bode like fundamental limitation for thecomplementary sensitivity function [2], which holds when adouble integrator is present in the return ratio, thus forcingthe transfer function relating successive distance errors tohave infinity norm greater than one.

It is known that such error propagation can be avoided ifthe distance from the leader is also used as part of the controllaw, provided the coupling with the leader is sufficientlystrong such that the infinity norm of the error propagationtransfer function is less than one. It has also been shown in[3] that for certain classes of systems and controllers, stringinstability can be avoided without the use of leader following,at the expense of successively more aggressive control lawswith linearly increasing gains.

In the paper we focus on symmetric bidirectional schemes,i.e. control laws where each vehicle tracks the average

distance from its neighbours. In its purely decentralizedform (no leader information) this is ill conditioned in thesense that the response to disturbances is not uniformlybounded with the size of the platoon [1], [4]. We show inthe paper that with arbitrarily weak coupling with the leader,the interconnection becomes scalable, i.e. the infinity normof the transfer function from disturbances to spacing errorsis uniformly bounded with the platoon size. In addition,we relax the homogeneity assumption on the dynamics andderive local conditions which, if satisfied, guarantee robuststability of an arbitrarily long heterogeneous interconnection.

The main idea is that even though more feedback is em-ployed in a bidirectional scheme, the underlying symmetryin the way information is communicated, makes it feasible toprovide guarantees for the global behaviour with conditionson the local interactions. This paradigm is also the centralproperty responsible for the scalability of Internet congestioncontrol models [5] where despite the heterogeneity, theunderlying symmetry dictated by TCP leads to robustnessresults for arbitrary networks.

The paper is structured as follows. We give in section IIthe problem formulation. In section III we discuss how leaderinformation makes a bidirectional scheme scalable, and insection IV we derive local stability conditions for heteroge-neous platoons. Examples are finally given to illustrate theresults derived.

II. PROBLEM FORMULATIONA. Notation

The field of real and complex numbers are denoted byR, C respectively. Rmn, Cmn are the m by n matri-ces with elements in the corresponding fields. C+ is theclosed right-half plane and R+ the closed set of positivereals. (M) denotes the spectrum of a matrix M CNN ,i(M) i = 1, . . . ,N its eigenvalues, (M) its spectral radiusand (M), (M) its maximum and minimum singular valuesrespectively. |M| is the elementwise absolute value of thematrix i.e. |[Mi j]| := [|Mi j|]. Co(S) denotes the convex hull ofa set S and diag(xi) the matrix with elements x1,x2, . . . on theleading diagonal and zeros elsewhere. The summation of twosets AC, BC is defined as A+B = {a+b : a A,b B}and we denote the square of a set P C as the set of thesquares of its elements i.e. P2 = {p2 : p P}.

The Numerical Range of a matrix M Cnn is theset N(M) := {vMv : v Cn,vv = 1} . Note the property

Proceedings of the 2007 American Control ConferenceMarriott Marquis Hotel at Times SquareNew York City, USA, July 11-13, 2007

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(M) N(M) (e.g. [6]). H is the Hardy space of transferfunctions of stable, linear, time-invariant, continuous-timesystems. The transfer functions in the paper are restricted tothe class of functions with continuous coprime factorizationsin jR{} (this is a necessary condition for the Nyquiststability criterion to be applicable).

B. Problem SetupWe adopt the same formulation as in [1]. We consider an

array of N +1 vehicles with positions denoted by xi(t),0i N, (x0(t) is the leader position), and their Laplace trans-form Xi(s). In this section we assume that all vehicles haveidentical dynamics with single-input-single-output transferfunction H(s) and use the same control law. The vehicleswant to maintain a fixed spacing distance > 0 and we alsoassume that they start from rest with x0(0) = 0,xi(0) =i .

In the Laplace domain the model of each vehicle is givenby

Xi(s) = H(s)(Ui(s)+Di(s))+xi(0)

sfor 1 i N (1)

where Ui(s) is the control input and Di(s) the input distur-bance. We also denote the spacing error as ei(t) = xi1(t)xi(t) and its Laplace transform

Ei(s) = Xi1Xi(s) s, 1 i N (2)

We consider control action of the form

Ui(s) = Kp(s)(Ei(s)Ei+1(s))+Kl(s)(

X0(s)Xi(s) is

)(3)

for 1 i < N ,

UN(s) = Kp(s)EN(s)+Kl(s)(

X0(s)XN(s) Ns

)(4)

where H(s) has a double pole1 at s = 0 and no other polesin C+ and Kp(s) H, Kl(s) H and have no pole/zerocancellations with H(s) in C+. Then from (1), (2), (3) weget

Ei(s) =H(s)Kp(s)[Ei1(s)+Ei+1(s)2Ei(s)]H(s)Kl(s)Ei(s)+H(s)[Di1(s)Di(s)]

for 1 i N ,Ei(s) =H(s)Kp(s)[Ei1(s)2Ei(s)]

H(s)Kl(s)Ei(s)+H(s)[Di1(s)Di(s)]for i = N , or in matrix form this can be written as

E(s) =H(s)Kp(s)LE(s)H(s)Kl(s)E(s)+H(s)MD(s)+1X0(s)

1The loopgains H(s)Kp(s), H(s)Kl(s) are assumed to have a double poleat s = 0 and this is assigned to H(s) as in [1], for convenience in thepresentation.

where

E(s) = [E1(s), . . . ,EN(s)]T , D(s) = [E1(s), . . . ,EN(s)]T

1 RN , 1 = [1 0 . . .0]T ,

L =

1 1 0 . . . 01 2 1 0 . . .

.

.

.

0 1 2 1 . . ..

.

.

.

.

.

.

.

.

.

.

. 0.

.

.

.

.

. 10 . . . 0 1 2

, (5)

M =

1 1

.

.

.

.

.

.

.

.

. 11

SoE(s) = Gx0e(s)X0(s)+Gde(s)D(s)

where

Gx0e(s) = [1+P(s)L]11, Gde(s) = P1(s)[1+P(s)L]1

P(s) =H(s)Kp(s)

1+H(s)Kl(s), P1(s) =

H(s)1+H(s)Kl(s)

III. SCALABILITY IN BIDIRECTIONAL CONTROL

It is shown in this section that information about theleader position can make Gde(s), Gx0e(s) uniformlybounded with the platoon size. In fact, it is shown that thiscan be achieved with arbitrarily weak coupling with theleader(Kl(s) arbitrarily small) for any robustly stabilizingbidirectional control Kp(s).

The following lemma is based on ideas in [7] (an analo-gous version also appears in [4]).

Lemma 1: An interconnection of N vehicles is stable if1

1+P(s) H (L), and P1(s) HProof: Since L is symmetric U RNN , U unitary,

such that L = UUT where = diag(i) , 1, . . .N are theeigenvalues of L. Now

[I +P(s)L]1 = U [I +P(s)]1UT = Udiag( 1

1+iP(s))

UT

So Gx0e(s) H. Since P1(s) H this also implies thatGde(s) H .Now let P := N{(L) : N = 1,2, . . .} and denote its closureby P. Note that a Gershgorin disc bound on the eigenvaluesof L [6] (this holds for all N), implies that P [0,4].

Lemma 2: Consider the interconnection in (1)-(3) and letKp(s), Kl(s) be such that

11+ H(s)Kp(s)1+H(s)Kl(s)

H P, H(s)1+H(s)Kl(s) H

then M1 > 0, M2 > 0 such that Gx0e(s) < M1,Gde(s) < M2 for all N {1,2, . . .} .

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Proof: From Lemma 1 the conditions in the The-orem imply that the interconnection is stable N. HenceGx0e,Gde H and for given L.

[I +P(s)L]12

= supR

12(I +P( j)L)

= supR

1mini i[(I +P( j)L)(I +P( j)L)]

= supR

1mini i[I +(P( j)+P( j))L+ |P( j)|2L2]

= supR

1mini[1+(P( j)+P( j))i(L)+ |P( j)|2i(L)2]

= supR

1mini |1+P( j)i(L)|2

The conditions in the Lemma imply

1 / {P( j) : {R}, P}=: FBy continuity of the frequency response on the compactifiedreal line, F is a compact set, hence > 0 such that f F |1+ f |> , therefore [I +P(s)L]1 1 . Sincealso P1(s) H both Gde(s), Gx0e(s) are uniformlybounded with N.

Remark 1: Uniform boundedness of Gx0e(s) is lostwhen there is no leader information (Kl(s) = 0) since|P( j)| as 0 and also its phase tends to pi . Hencefor sufficiently small P( j) can be arbitrarily close to1 (note also analysis in [4]).

The Theorem below shows that if the bidirectional controlKp(s) is such that there is a finite stability margin then uni-form boundedness of Gde(s), Gx0e(s) can be achievedwith arbitrarily weak coupling with the leader.

Theorem 1: Given Kp(s) H such that

> 0 with 11+H(s)Kp(s)

H (P+(0,))(6)

then > 0 Kl(s) H, M1 > 0, M2 > 0 such thatKl(s) < and Gx0e(s) < M1, Gde(s) < M2 for allN {1,2, . . .} .

Remark 2: 11+H(s)Kp(s) H P is a necessarycondition for stability of the platoon with no leader infor-mation (Kl(s) = 0). Therefore condition (6) in the Theoremrequires an arbitrarily small stability margin. This is easy toverify by noting that P [0,4].

Proof: Choose Kl(s) = kKp(s) s.t. Kp(s)< and1

1+ H(s)Kp(s)1+kH(s)Kp(s)=

1+ kH(s)Kp(s)1+( + k)H(s)Kp(s)

H P,

H(s)1+ kH(s)Kp(s)

H

Such a k always exists from the following argument. Thesmallest eigenvalue of L , 1(L) 0 as N [4]. So bychoosing k = 1(L) k can be made arbitrarily small for alarge enough N. A suitable k = 1(L) is one that satisfies0 < k < min

{, Kp(s)

}.

The proof then follows in analogy with Lemma 2 toshow that Gde(s), Gx0e(s) are uniformly boundedwith N i.e. M1 > 0, M2 > 0 such that Gx0e(s) < M1,Gde(s) < M2 for all N {1,2, . . .} .IV. ROBUST STABILITY OF HETEROGENEOUS PLATOONS

In this section we relax the homogeneity assumption onthe vehicle dynamics and control laws in the platoon. In abidirectional scheme this adds additional complication in theanalysis since the underlying graph is strongly connected i.e.each vehicle (apart from the leader) is indirectly coupled withsome feedback loop with all other vehicles. We show belowthat the symmetry of the information flow makes it possibleto derive local conditions which, if satisfied, guarantee robuststability of an arbitrarily long heterogeneous interconnection.

The problem formulation is the same as in section III withthe dynamics of the ith vehicle Hi(s) and the control lawsKip(s),Kil (s) not being necessarily the same. Note, however,that because of this heterogeneity, spacing error can nolonger be written as a function of only other spacing errorsand the leader position. In the Laplace domain the positionof the ith vehicle is given by

Xi(s) = Hi(s)(Ui(s)+Di(s))+xi(0)

sfor 1 i N (7)

and the control input

Ui(s) = Kip(s)(Ei(s)Ei+1(s))+Kil (s)(

X0(s)Xi(s) is

)(8)

for 1 i < N ,

UN(s) = KNp (s)EN(s)+KNl (s)(

X0(s)XN(s) Ns

)(9)

In matrix form this can be written as

X(s) =diag(Pi(s))(

LX(s)1X0(s)+N s

)+diag(Pi1(s))D(s)+diag(Pi2(s))X0(s)

where

Pi(s) =Hi(s)Kip(s)

1+Hi(s)Kil (s), Pi1(s) =

Hi(s)1+Hi(s)Kil (s)

,

Pi2(s) =Hi(s)Kil (s)

1+Hi(s)Kil (s),

X0(s) =[

X0(s) s, . . . ,X0(s) i

s, . . . ,X0(s) N

s

]TL is as in (5) with L11 = 2, LNN = 1 and N = [0 . . . 0 1]T .So

X(s) = [I +diag(Pi(s))L]1[P1(s)1X0(s)PN(s)N

s

+diag(Pi1(s))D(s)+diag(Pi2(s))X0(s)]

Theorem 2 (based on Lemma 3) gives local conditions thatguarantee stability of an arbitrarily long interconnection. Wefirst denote di the in-degree of each vehicle, i.e. di = 2 for1 i < N, dN = 1 and let Din = diag(di).

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Lemma 3: ([8]) Let Q Cnn, Q = Q 0 and G =diag(gi),gi C,i {1, . . .n}. Then

(GQ) (Q)Co(0{gi : i {1, . . . ,n}})Proof: {0}(GQ) = {0}(Q1/2GQ1/2). But

(Q1/2GQ1/2) N(Q1/2GQ1/2):= {vQ1/2GQ1/2v : v Cn,v= 1} (Q){wGw : w Cn,w 1}

= (Q){n

i=1|wi|2gi : wi C,

n

i=1|wi|2 1}

= (Q)Co(0{gi : i = 1, . . . ,n})

Theorem 2: The interconnection in (7)(9) is stable for allN if Pi1(s) H i and

1 / R+{

Co({2diPi( j) : i = 1, . . . ,N}0)} (10)

Remark 3: The conditions in the theorem are decentral-ized. This can be seen by means of a duality argument.Requiring the convex hull of the frequency responses andzero not to include the point 1 is equivalent to requiringeach of the frequency responses to lie to the right of ahyperplane through the point 1. This hyperplane can befrequency dependent since we are taking the union of theconvex hulls over frequency.Also the distance of each frequency response from thehyperplane gives a measure of robustness i.e. it is guaranteedthat the system will remain stable for an additive perturba-tion (s) H in Pi(s), with H norm smaller than thiscorresponding distance.

Proof: (of Theorem 2) It is sufficient to show that theeigenloci of the return ratio diag(Pi(s))L do not encircle thepoint 1. Now diag(Pi(s))L is similar to diag(2diPi(s))Mwhere M = (2Din)1/2L(2Din)1/2 . A Gershgorin discbound on the eigenvalues of (2Din)1L shows that M 0,(M) 1. Therefore since also M = MT , we have fromLemma 3

(diag(2diPi( j))L)Co({2diPi( j) : i = 1, . . . ,N}0)

Now

R{

Co({2diPi( j) : i = 1, . . . ,N}0)} (11)

is a set which is star shaped with respect to zero2. Thereforethe fact that it does not include the point 1 also ensures thatit does not encircle it. Note also that due to the symmetry ofthe Nyquist plot about the real axis, it is sufficient to checkthat 1 is not included in the set in (11) with the union takenonly over positive frequencies, as in the Theorem.

Potentially less conservative stability certificates can beobtained by deriving conditions that involve products offrequency responses of neighbouring dynamics. This can be

2A set PC is star shaped with respect to point x0 P if for any p Pthe line segment between p and x0 is in P, i.e. {tx0 +(1t)p : t [0,1]}P

achieved by rewriting the return ratio as shown below

Xi(s) =2HiKip(s)

1+Hi(2Kip(s)+Kil (s)) Pi3

12(Xi1 +Xi+1)

+Hi(s)

1+Hi(2Kip(s)+Kil (s)) Pi5

Di(s)+Pi5(s)Kil (s)

Pi4

(X0 i )

for 1 i < N, and

XN(s) =HNKNp

1+HN(KNp +KNp ) PN3

(XN1

s

)

+HN(s)

1+HN(KNp (s)+KNl (s)) PN5

DN(s)+KNl (s)PN5 (s)

PN4

(X0N )

or in matrix form

X(s) = diag(Pi3(s))(Din)1[AX(s)+1X0(s)N

s

]+diag(Pi4(s))X0(s)+diag(Pi5(s))D(s)

where

A =

0 1 0 . . . 0

1 0 1.

.

.

0.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. 10 . . . 1 0

Note that A is the adjacency matrix of the underlying graph,i.e. Ai j = 1 if vehicle i communicates with vehicle j and 0otherwise. Theorem 3 makes use of Lemma 4 which is basedon the S-hull, a relaxation of the convex hull of a set in C.

Definition 1 (S-hull): Let P C. The S-hull of set P isdefined as

S(P) := (Co(

P))2 where

P := {x : x2 P}

0.5 0 0.5 1 1.5 2 2.5 31

0.5

0

0.5

1

Re[z]

Im[z]

Fig. 1. The convex hull and Shull of four points (denoted with circles).

It can be shown [9] that the S-hull S(P) of a set P Cis always a convex set that includes zero (see example in

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figure 1). Its importance lies in the fact that it can be used tobound the spectrum of a product of matrices with a particularstructure by means of non zero elements of those matrices.

Lemma 4: ([9]) Let R Cmn satisfy (|R|T |R|) 1, andG = diag(g1, . . . ,gn), F = diag( f1, . . . , fm), gi, f j C i, jthen

(FRGR)Co({ fiS({gk : Rik 6= 0}) : i = 1, . . . ,m})Theorem 3: The interconnection in (7)(9) is stable if

Pi5(s) H i and

1 / R+Co(i:i{1,...,N},i oddS({Pi3( j)P j3 ( j) : Ai j 6= 0})

)(12)

Proof: Note first that the underlying graph is bipartitesince vehicles with an odd(even) index communicate onlywith vehicles with an even(odd) index. So by permutingthe rows of A, this can be brought to an antidiagonal formA =

[0 A1

AT1 0

]where A1 is 0 1 matrix with its rows and

columns indexed by the odd and even vehicles respectively.Similarly let diag(Pi3(s)) = diag(G1(s),G2(s)) where G1(s) =diag({Pi3(s) : i odd}), G2(s) = diag({Pi3(s) : i even}) andDin = diag(D1in,D2in), D1in(s) = diag({di : i odd}), D2in(s) =diag({di : i even}). Now the return ratio is

L(s) :=diag(Pi3(s))(Din)1A

=

0 G1(s)(D1in)1 G1(s)

A1

G2(s)(D2in)1 G2(s)

AT1 0

and det(IL(s)) = det(IG1(s)A1G2(s)AT1 ) hence

1 / (kG1( j)AG2( j)AT ) k [0,1] 1 / (kL( j)) k [0,1]

Note that G1(s)A1G2(s)AT1 is similar to G1(s)RG2(s)RTwhere R = (D1in)1/2A1(D2in)1/2 and (|R||R|T ) 1 since

(|R||R|T ) = ((D1in)1A1(D2in)1AT1 ) (D1in)1A1(D2in)1AT1 1

Hence from Lemma 4, the bounding set in (12) is a boundfor the spectrum of G1( j)AG2( j)AT R+ and is alsostar shaped with respect to zero as the S-hull always includesthe point 0. So 1 /(kL( j)) k [0,1], R. Thereforethe eigenloci do not encircle the point 1 and the system isstable from the multivariable Nyquist criterion [10].

Corollary 1: The interconnection in (7)(9) is stable ifPi5(s) H i and

1 /

R+Co

( i:i{1,...,N},i even

S({Pi3( j)P j3 ( j) : Ai j 6= 0}))

Proof: The proof is the same as in Theorem 3 but withthe rows and columns of A1 indexed by the even and oddvehicles respectively.

102 101 100 1010

5

10

15

20k=0.1

N=2

N=6

N=50

(rad/s)

(G

de(

j))

102 101 100 1010

0.5

1

1.5

2

2.5k=0.5

N=2

N=6 N=50

(rad/s)

(Gde

(j

))

102 101 100 1010

0.2

0.4

0.6

0.8

1k=1

N=2

N=6 N=50

(rad/s)

(G

de(

j))

Fig. 2. (Gde( j)) for a control law Kl(s) = kKp(s) for k = 0.1,0.5,1and with N = 2,6,10, . . . ,50.

V. EXAMPLESFigure 2 shows an example of a bidirectional control

scheme with

H(s) =1

s2(0.1s+1) Kp(s) =2s+1

0.05s+1 (13)

and Kl(s) = kKp(s). As expected, Gde is uniformlybounded with the size of platoon for all k > 0. In additionbetter disturbance rejection properties are observed with astronger coupling with the leader position (larger k). If,on the other hand, only predecessor following was usedwith the same controller, then the transfer function betweensuccessive spacing errors is T (s) = H(s)Kp(s)1+H(s)Kp(s)(1+k) and erroramplification is avoided (T (s) < 1) only for k > 0.19.Figure 3 gives an example that illustrates how the convexhull condition in Theorem 2 can be applied to guarantee

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2 1 0 1 2 3 4 56

5

4

3

2

1

0

1

Re[4Pi( j)]

Im[4

Pi(

j)]

2 1 0 1 2 3 4 56

5

4

3

2

1

0

1

Re[4Pi( j)]

Im[4

Pi(

j)]

Fig. 3. R+{

Co({4Pi( j) : i = 1, . . . ,5}0)} with = 13 , 12 ,1,2,3

for each i respectively. In plot on top Kp(s) = 2s+10.05s+1 and in second plotKp(s) = 12

2s+10.05s+1 .

stability of a heterogeneous interconnection. We considervehicles with varying actuation lag i.e.

H(s) =1

s2(0.1s+1) , > 0

We initially use the same controller as in (13) withKp(s) = Kl(s). The top plot in figure 3 (solid lines)shows the Nyquist plot of 4Pi( j) for R+, i =1,2, . . . ,5 where Pi( j) = Hi( j)K

ip( j)

1+Hi( j)Kip( j) and =13 ,

12 ,1,2,3

for each Hi(s) respectively. The thinner line indicatesCo

({4Pi( j) : i = 1, . . . ,5}0) at each frequency. Note thatthe union of the convex hulls does not include the point 1so the interconnection is stable from Theorem 2. In fact, amore informative design would be to reduce the gain of thecontroller as the lag increases and also add the phase leadat lower frequencies. The second plot in figure 3 illustratessuch an approach with Kp(s) = 12

2s+10.05s+1 . Note that the five

Nyquist plots are on top of one another (though they areshifted in frequency) and much better stability margins canbe guaranteed since the distance of their convex hull fromthe point 1 is greater. As also mentioned in Remark 3,stability of an arbitrarily long heterogeneous interconnectioncan be guaranteed by choosing a line through the point 1and ensuring that the each of the 4Pi( j) does not intersectthis line.

Figure 4 shows how Theorem 3 can be used to cer-tify stability in the same example (we use Kp(s) =12

2s+10.05s+1 and Kl(s) = 0.3Kp(s)). The plot illustrates

1 0.5 0 0.5 11.2

1

0.8

0.6

0.4

0.2

0

0.2

0.4

Re[z]

Im[z

]

Fig. 4. P33 ( j)P43 ( j), R+ (solid line) and S({P33 ( j)P j3 ( j) : A3 j 6=0}), R+ .

R+S({Pi3( j)P j3 ( j) : Ai j 6= 0}) for i = 3 and note thatthis does not include the point 1. For stability it is sufficientthat the for all odd i, the set above does not intersect a linethrough the point 1 (as in Remark 3) .

VI. CONCLUSIONSIt has been shown in the paper that arbitrarily weak

coupling with the leader makes a symmetric bidirectionalscheme for the control of a platoon of vehicles scalable,i.e. the response to disturbances is uniformly bounded withthe size of the platoon. In addition the symmetry of theinterconnections can be exploited to relax the homogeneityassumption on the dynamics and derive local conditions,which, if satisfied, guarantee stability of an arbitrarily longheterogeneous interconnection. This is an example that il-lustrates how by incorporating symmetries in underlyingprotocols can lead to large scale heterogeneous networkswhich are robust and scalable.

REFERENCES[1] P. Seiler, A. Pant, and K. Hedrick, Disturbance propagation in vehicle

strings, IEEE Transactions on Automatic Control, vol. 49, no. 10, pp.18351841, October 2004.

[2] R. Middletton and G. Goodwin, Digital Control and Estimation: aUnified approach. Upper Saddle River, NJ: Prentice-Hall, 1990.

[3] M. E. Khatir and E. J. Davidson, Bounded stability and eventualstring stability of a large platoon of vehicles using non-identicalcontrollers, in Proceedings of IEEE Conference on Decision andControl, 2004.

[4] P. Barooah and J. P. Hespanha, Error amplification and disturbancepropagation in vehicle strings with decentralized linear control, inProceedings of IEEE Conference on Decision and Control, 2005.

[5] R. Srikant, The mathematics of Internet congestion control.Birkhauser, 2004.

[6] R.A.Horn and C. R. Johnson, Matrix Analysis. New York: CambridgeUniversity Press, 1999.

[7] A. Fax and R. M. Murray, Information flow and cooperative control ofvehicle formations, IEEE Transactions on Automatic Control, vol. 49,pp. 14651476, September 2004.

[8] G. Vinnicombe, On the stability of end-to-end congestion control forthe Internet, Cambridge University Engineering Department, Tech.Rep. CUED/F-INFENG/TR.398, 2000.

[9] I. Lestas and G. Vinnicombe, Scalable decentralized robust stabilitycertificates for networks of interconnected heterogeneous dynamicalsystems, IEEE Transactions on Automatic Control, vol. 51, no. 10,pp. 16131625, October 2006.

[10] C. A. Desoer and Y. T. Yang, On the generalized Nyquist stabilitycriterion, IEEE Transactions on Automatic Control, vol. 25, pp. 187196, 1980.

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